Trigonometric Functions - IB Math AA Question Bank

Question 1

SL • Paper 1

Easy

Non Calculator

The point $P$ lies on the unit circle and corresponds to the angle $\theta$ , where $\pi<\theta<\dfrac{3\pi}{2}$ . Given that $\tan\theta=\sqrt{3}$ :

A

Find the value of $\theta$ .

[2]

Write your answer here...

B

Write down the coordinates of $P$ .

[2]

Write your answer here...

C

line through the origin makes an angle $2\pi-\theta$ with the positive $x$ -axis. Find the gradient of the line.

[1]

Write your answer here...

0

Question 2

SL • Paper 1

Easy

Non Calculator

Let $x$ be an angle such that $\dfrac{\pi}{2}<x<\pi$ and $\cos x=-\dfrac{3}{5}$ .

A

Find $\sin x$ .

[2]

Write your answer here...

B

Find $\cos 2x$ .

[1]

Write your answer here...

C

Find $\sin 2x$ .

[2]

Write your answer here...

0

Question 3

SL • Paper 1

Easy

Non Calculator

A

Solve $2\sin\left(x-\dfrac{\pi}{4}\right)=1$ for $0\le x\le 2\pi$ .

[4]

Write your answer here...

0

Question 4

SL • Paper 2

Easy

Calculator Permitted

A line $l$ passes through the origin and makes an angle $\theta$ with the positive $x$ -axis, where $\pi<\theta<\dfrac{3\pi}{2}$ . A point $P(-4,k)$ lies on $l$ . Given that $\sin\theta=-\dfrac{5}{13}$ , answer the following.

A

Find $\cos\theta$ .

[1]

Write your answer here...

B

Find the exact value of $k$ .

[2]

Write your answer here...

C

Find the exact value of $\cos 2\theta$ .

[2]

Write your answer here...

0

Question 5

SL • Paper 2

Easy

Calculator Permitted

Let $x$ be an angle such that $\dfrac{\pi}{2}<x<\pi$ and $\cos x=-0.28$ .

A

Find $\sin x$ .

[2]

Write your answer here...

B

Find $\sin 2x$ and $\cos 2x$ .

[3]

Write your answer here...

0

Question 6

SL • Paper 1

Medium

Non Calculator

In triangle $ABC$ , $A=30^\circ$ , $a=5$ and $b=5\sqrt{2}$ , where $a$ and $b$ are opposite $A$ and $B$ respectively.

A

Find the possible values of angle $C$ .

[5]

Write your answer here...

0

Question 7

SL • Paper 1

Medium

Non Calculator

A periodic quantity is modelled by $f(t)=a\sin(bt)+d$ , where $b>0$ . The maximum value of $f$ is $10$ , the minimum value is $2$ , and the period is $\pi$ . Also, for small positive values of $t$ , $f(t)>f(0)$ .

A

Find the amplitude and the value of $d$ .

[2]

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B

Find the value of $b$ .

[2]

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C

Determine the values of $a$ , $b$ and $d$ .

[2]

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0

Question 8

SL • Paper 1

Medium

Non Calculator

A

Solve $2\cos^2 x-\cos x-1=0$ for $0\le x\le 2\pi$ .

[5]

Write your answer here...

0

Question 9

HL • Paper 1

Medium

Non Calculator

Let $\pi<\theta<\dfrac{3\pi}{2}$ and $\tan\theta=2$ .

A

Find $\sec\theta$ .

[3]

Write your answer here...

B

Find $\csc\theta$ .

[2]

Write your answer here...

0

Question 10

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=\arccos(2x-1)$ .

A

Find the domain of $f$ .

[2]

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B

Solve $f(x)=\dfrac{2\pi}{3}$ .

[3]

Write your answer here...

0

Question 11

HL • Paper 1

Medium

Non Calculator

Let $0<\theta<\dfrac{\pi}{4}$ and $\tan\theta=\dfrac{1}{3}$ .

A

Find $\tan 2\theta$ .

[3]

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B

Hence find $\tan\left(2\theta+\dfrac{\pi}{4}\right)$ .

[2]

Write your answer here...

0

Question 12

HL • Paper 1

Medium

Non Calculator

Consider the equation $\sin(\pi-x)=\cos(-x)$ for $-\pi\le x\le \pi$ .

A

Use symmetry properties to rewrite the equation in the form $\sin x=\cos x$ .

[2]

Write your answer here...

B

Hence solve the equation for $-\pi\le x\le \pi$ .

[2]

Write your answer here...

0

Question 13

SL • Paper 2

Medium

Calculator Permitted

Consider the equation

2\sin^2x-3\cos x=0

for $0\le x\le 2\pi$ .

A

Show that the equation can be written as $2\cos^2x+3\cos x-2=0$ .

[1]

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B

Hence solve the equation.

[3]

Write your answer here...

0

Question 14

HL • Paper 2

Medium

Calculator Permitted

Let $\theta$ be an angle such that $\dfrac{\pi}{2}<\theta<\pi$ and $\tan\theta=-\dfrac{12}{5}$ .

A

Find the exact values of $\sec\theta$ and $\csc\theta$ .

[3]

Write your answer here...

B

Solve $\sec^2x-3\tan x=5$ for $0\le x<2\pi$ .

[3]

Write your answer here...

0

Question 15

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\arccos(2x-1)

A

State the domain and range of $f$ .

[2]

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B

Solve exactly $f(x)=2\arctan\left(\dfrac{3}{4}\right)$ .

[3]

Write your answer here...

0

Question 16

HL • Paper 2

Medium

Calculator Permitted

The graph of $y=2\csc(x-0.4)+1$ is considered for $0\le x\le 2\pi$ .

Graph of y = 2 csc(x - 0.4) + 1 for 0 ≤ x ≤ 2π.

A

Find the equations of the vertical asymptotes in this interval.

[2]

Write your answer here...

B

Solve $2\csc(x-0.4)+1=5$ in this interval.

[2]

Write your answer here...

C

State the range of $y=2\csc(x-0.4)+1$ .

[1]

Write your answer here...

0

Question 17

HL • Paper 1

Medium

Non Calculator

This question uses the compound angle identities.

A

Show that $\sin\dfrac{\pi}{12}=\dfrac{\sqrt{6}-\sqrt{2}}{4}$ .

[3]

Write your answer here...

B

Hence solve $\sin\left(x+\dfrac{\pi}{12}\right)=\dfrac{\sqrt{6}-\sqrt{2}}{4}$ for $0\le x\le 2\pi$ .

[3]

Write your answer here...

0

Question 18

HL • Paper 1

Medium

Non Calculator

A

Solve $\sec x-\tan x=\dfrac{1}{3}$ for $0\le x<\dfrac{\pi}{2}$ .

[6]

Write your answer here...

0

Question 19

SL • Paper 2

Medium

Calculator Permitted

In triangle $ABC$ , $AB=9\ \text{cm}$ , $AC=7\ \text{cm}$ and $\angle ABC=0.620$ radians. This information gives two possible triangles.

Redraw a geometrically accurate non-right triangle showing one valid configuration only. Place $B$ and $A$ so that $AB=9\ \text{cm}$; from $B$, draw the ray $BC$ so that $\angle ABC=0.620$ radians, and choose the nearer intersection of this ray with the circle centered at $A$ of radius $7\ \text{cm}$, giving $AC=7\ \text{cm}$. Label $A$, $B$, $C$, $AB=9\ \text{cm}$, $AC=7\ \text{cm}$, and the angle at $B$. The angle at $B$ should appear approximately $35.5^\circ$, not $61^\circ$. Do not show the second possible position explicitly, and draw the figure to scale as far as practical.

A

Find the two possible values of $\angle ACB$ .

[3]

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B

Hence find the two possible areas of triangle $ABC$ .

[3]

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0

Question 20

SL • Paper 2

Medium

Calculator Permitted

The height $h$ metres of a floating marker above the sea bed is modelled by

h(t)=a\sin(b(t-c))+d

where $t$ is the time in hours after midnight and $a>0$ . The maximum height is $5.8$ metres at $t=2$ , the minimum height is $1.4$ metres at $t=8$ , and the motion has period $12$ hours, with $b>0$ and $-6<c\le 6$ .

Sinusoidal marker height over one 12-hour cycle.

A

Find the values of $a$ , $b$ and $d$ .

[3]

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B

Find the value of $c$ .

[2]

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C

Find the first time after midnight when the height is $4.5$ metres.

[2]

Write your answer here...

0

Question 21

SL • Paper 2

Medium

Calculator Permitted

The equation

2\tan(3(x-1))=1

is to be solved for $0\le x\le 3$ .

A

Let $u=3(x-1)$ . Find the corresponding interval for $u$ .

[1]

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B

Hence solve the equation for $x$ .

[4]

Write your answer here...

0

Question 22

HL • Paper 2

Medium

Calculator Permitted

Let $A$ and $B$ be acute angles such that $\tan A=2$ and $\tan B=\dfrac{1}{3}$ .

A

Find the exact value of $\tan(A+B)$ .

[2]

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B

Find $A+B$ , giving your answer to $3$ significant figures.

[1]

Write your answer here...

C

Hence solve $\tan(2x)=7$ for $0\le x\le \pi$ .

[3]

Write your answer here...

0

Question 23

HL • Paper 2

Medium

Calculator Permitted

For $0\le x\le 2\pi$ , define

f(x)=\sin x+\sin(\pi-x)+\cos(\pi+x)

A

Use symmetry properties of trigonometric functions to simplify $f(x)$ .

[2]

Write your answer here...

B

Write $f(x)$ in the form $R\sin(x-\alpha)$ , where $R>0$ and $0<\alpha<\dfrac{\pi}{2}$ .

[2]

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C

Solve $f(x)=1$ .

[2]

Write your answer here...

0

Question 24

SL • Paper 1

Medium

Non Calculator

Let $\theta$ be an angle such that $\dfrac{\pi}{2}<\theta<\pi$ and $\cos\theta=-\dfrac{5}{13}$ . A point $P$ on the unit circle corresponds to the angle $\theta$ .

A

I.

Find $\sin\theta$ .

[2]

Write your answer here...

II.

Hence find $\tan\theta$ .

[2]

Write your answer here...

B

I.

line $l$ passes through the origin and is perpendicular to $OP$ . Find the gradient of $l$ .

[2]

Write your answer here...

II.

Write down an equation of $l$ .

[1]

Write your answer here...

C

The point $P$ is reflected in the line $y=x$ . The image is $Q$ . Determine the angle $\alpha$ , $0\leq \alpha<2\pi$ , corresponding to $Q$ on the unit circle, in terms of $\theta$ . Hence find $\tan\alpha$ .

[3]

Write your answer here...

0

Question 25

SL • Paper 1

Medium

Non Calculator

In triangle $ABC$ , side lengths $a$ , $b$ and $c$ are opposite angles $A$ , $B$ and $C$ respectively. It is given that $A=45^\circ$ , $a=4$ and $b=2\sqrt{6}$ .

A

I.

Use the sine rule to show that $\sin B=\dfrac{\sqrt{3}}{2}$ .

[2]

Write your answer here...

II.

Find the two possible values of $B$ .

[2]

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B

I.

Find the corresponding two possible values of $C$ .

[2]

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II.

Find the two possible exact values of $c$ .

[2]

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C

Find the two possible exact areas of triangle $ABC$ .

[3]

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0

Question 26

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by

f(x)=3-2\sin\left(2x+\frac{\pi}{3}\right),\quad 0\leq x\leq \pi

A

I.

State the amplitude and period of $f$ .

[2]

Write your answer here...

II.

Describe the horizontal and vertical translations from $y=-2\sin(2x)$ to $y=f(x)$ .

[2]

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B

I.

Find the maximum and minimum values of $f$ .

[2]

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II.

Find the values of $x$ in the given interval for which $f(x)=5$ .

[2]

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C

Solve $f(x)=4$ for $0\leq x\leq \pi$ .

[3]

Write your answer here...

0

Question 27

SL • Paper 2

Medium

Calculator Permitted

The brightness $B$ of a rotating warning light, in arbitrary units, is modelled by

B(t)=a\sin(b(t-c))+d,\quad 0\leq t\leq 16

where $t$ is the time in seconds after the light is switched on and $a>0$ . The brightness has a maximum value of $72$ at $t=1.5$ and a minimum value of $18$ at $t=5.5$ . The motion is periodic.

Brightness of a warning light over two cycles.

A

I.

Find the values of $a$ and $d$ .

[2]

Write your answer here...

II.

Find the value of $b$ , taking $b>0$ .

[2]

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III.

Determine one possible value of $c$ for this choice of $b$ .

[2]

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B

Using your model, find all times in the interval $0\leq t\leq 16$ when $B(t)=60$ . If you did not obtain a model in part (a), use $B(t)=27\sin\left(\dfrac{\pi}{4}(t+0.5)\right)+45$ .

[3]

Write your answer here...

C

Determine the total length of time, in seconds, during $0\leq t\leq 16$ for which the brightness is greater than $60$ .

[3]

Write your answer here...

0

Question 28

SL • Paper 2

Medium

Calculator Permitted

In triangle $ABC$ , the side lengths opposite $A$ , $B$ and $C$ are $a$ , $b$ and $c$ respectively. It is given that $A=0.650$ radians, $a=10.0\ \text{cm}$ and $b=13.0\ \text{cm}$ .

A non-right triangle labelled $A$, $B$, $C$, with side $a$ opposite angle $A$ and side $b$ opposite angle $B$. The diagram should indicate that two different positions of vertex $B$ may be possible for the same given data.

A

I.

Use the sine rule to find the two possible values of angle $B$ .

[4]

Write your answer here...

B

I.

For each possible triangle, find the corresponding value of $C$ .

[2]

Write your answer here...

II.

For each possible triangle, find the corresponding length of $c$ .

[2]

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C

design condition states that the area of triangle $ABC$ must exceed $20\ \text{cm}^2$ . Determine which of the possible triangles satisfies this condition.

[2]

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0

Question 29

SL • Paper 2

Medium

Calculator Permitted

A machine component moves vertically. Its height $h$ cm above a fixed level is modelled by

h(x)=a\cos(b(x-c))+d

where $x$ is measured in seconds and $b>0$ . The graph of $h$ has consecutive maximum points at $(2,9)$ and $(12,9)$ , and a minimum point at $(7,1)$ .

A

I.

Find the amplitude and the value of $d$ .

[2]

Write your answer here...

II.

Find the value of $b$ .

[2]

Write your answer here...

III.

Find a possible model for $h(x)$ .

[3]

Write your answer here...

B

second component has height $g(x)=h(x-3)+2$ . Describe the transformation from the graph of $h$ to the graph of $g$ .

[2]

Write your answer here...

C

Find the first time after $x=0$ when the height of the second component is $8$ cm. If you did not obtain a model in part (a), use $h(x)=4\cos\left(\dfrac{\pi}{5}(x-2)\right)+5$ .

[3]

Write your answer here...

0

Question 30

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

\arctan x+\arctan(2x)=\frac{\pi}{4}

A

Show that any solution must satisfy $2x^2+3x-1=0$ .

[2]

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B

Hence find the exact solution of the original equation.

[3]

Write your answer here...

0

Question 31

SL • Paper 1

Hard

Non Calculator

The height $h$ metres of a point on a rotating wheel above the ground is modelled by

h(t)=a\cos(b(t-c))+d

where $t$ is the time in seconds, $a>0$ and $b>0$ . The maximum height is $14$ metres at $t=1$ , and the minimum height is $2$ metres at $t=5$ . The maximum at $t=1$ and the minimum at $t=5$ are consecutive extrema.

A

I.

Find $a$ and $d$ .

[2]

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II.

Find $b$ .

[2]

Write your answer here...

III.

Find a possible value of $c$ .

[1]

Write your answer here...

B

Hence write down a model for $h(t)$ .

[2]

Write your answer here...

C

I.

Find the first time after $t=1$ when $h(t)=8+3\sqrt2$ .

[3]

Write your answer here...

II.

Determine the values of $t$ in $1\leq t\leq 9$ for which $h(t)=5$ .

[2]

Write your answer here...

0

Question 32

SL • Paper 1

Hard

Non Calculator

Consider the equation

2\sin x=\cos 2x

for $-\pi\leq x\leq \pi$ .

A

I.

Show that the equation can be written as $2\sin^2x+2\sin x-1=0$ .

[3]

Write your answer here...

II.

Hence find the possible values of $\sin x$ .

[2]

Write your answer here...

B

Let $\alpha=\arcsin\left(\dfrac{\sqrt3-1}{2}\right)$ , where $0<\alpha<\dfrac{\pi}{2}$ . Solve the equation for $-\pi\leq x\leq \pi$ .

[3]

Write your answer here...

C

I.

Find the exact value of $\cos 2x$ for each solution.

[2]

Write your answer here...

II.

Determine whether the equation has any solution in $-\pi\leq x<0$ . Give a reason.

[2]

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0

Question 33

SL • Paper 1

Hard

Non Calculator

Consider the equation

2\tan\left(2x-\frac{\pi}{4}\right)=2

where $0\leq x\leq 2\pi$ .

A

I.

Let $u=2x-\dfrac{\pi}{4}$ . Find the corresponding interval for $u$ .

[2]

Write your answer here...

II.

Solve the equation for $u$ in this interval.

[2]

Write your answer here...

B

Hence solve the original equation for $x$ .

[3]

Write your answer here...

C

I.

Find the values of $x$ in $0\leq x\leq 2\pi$ for which $\tan\left(2x-\dfrac{\pi}{4}\right)$ is undefined.

[3]

Write your answer here...

II.

State the number of branches of the graph of $y=\tan\left(2x-\dfrac{\pi}{4}\right)$ in $0\leq x\leq 2\pi$ .

[2]

Write your answer here...

0

Question 34

HL • Paper 1

Hard

Non Calculator

Let $0\leq x<2\pi$ . Consider the equation

\sec x+\tan x=3

A

AI.

Show that $(\sec x+\tan x)(\sec x-\tan x)=1$ .

[2]

Write your answer here...

AII.

Hence find $\sec x-\tan x$ .

[2]

Write your answer here...

B

BI.

Find $\sec x$ and $\tan x$ .

[2]

Write your answer here...

BII.

Hence find $\sin x$ and $\cos x$ .

[2]

Write your answer here...

C

Solve $\sec x+\tan x=3$ for $0\leq x<2\pi$ .

[4]

Write your answer here...

0

Question 35

HL • Paper 1

Hard

Non Calculator

The function $f$ is defined by

f(x)=\arcsin(2x-1)

A

AI.

Find the domain of $f$ .

[2]

Write your answer here...

AII.

State the range of $f$ .

[2]

Write your answer here...

B

BI.

Find $f\left(\dfrac{1}{4}\right)$ .

[2]

Write your answer here...

BII.

Solve $f(x)=\dfrac{\pi}{3}$ .

[2]

Write your answer here...

C

Let $g(x)=\arccos(2x-1)$ on the same domain as $f$ . Show that $f(x)+g(x)=\dfrac{\pi}{2}$ for $0\leq x\leq1$ .

[4]

Write your answer here...

0

Question 36

SL • Paper 2

Hard

Calculator Permitted

Consider the function

f(x)=2\sin x-\cos 2x,\quad 0\leq x\leq 2\pi

Curve of $f(x)=2\sin x-\cos 2x$ with a horizontal line at $y=1.5$.

A

I.

Show that $f(x)=2\sin^2x+2\sin x-1$ .

[2]

Write your answer here...

B

I.

Hence solve $f(x)=0$ for $0\leq x\leq 2\pi$ .

[4]

Write your answer here...

C

Use a GDC to find the maximum and minimum values of $f$ on $0\leq x\leq 2\pi$ .

[3]

Write your answer here...

D

Find the set of values of $x$ for which $f(x)>1.5$ .

[2]

Write your answer here...

0

Question 37

SL • Paper 2

Hard

Calculator Permitted

A narrow laser beam is emitted from the origin in the coordinate plane. At time $t$ seconds, the beam makes an angle

\theta=0.25t+0.300

with the positive $x$ -axis. A vertical screen is placed along the line $x=4$ . For times before the first parallel position, the point where the beam meets the screen has coordinates $(4,y)$ .

A coordinate diagram with the origin, a ray making angle $\theta$ with the positive $x$-axis, and a vertical screen at $x=4$. The intersection point $(4,y)$ should be labelled.

A

I.

Show that $y=4\tan(0.25t+0.300)$ .

[2]

Write your answer here...

II.

Find the first time at which the beam is parallel to the screen. Give your answer to three significant figures.

[2]

Write your answer here...

B

Find the time in the interval $0\leq t<\dfrac{\dfrac{\pi}{2}-0.300}{0.25}$ when $y=6$ .

[3]

Write your answer here...

C

At the time found in part (b), find the distance from the origin to the point where the beam meets the screen.

[2]

Write your answer here...

D

State why the model cannot give a finite value of $y$ at the first parallel time, $t=\dfrac{\dfrac{\pi}{2}-0.300}{0.25}$ (which is $5.08$ s to three significant figures).

[1]

Write your answer here...

0

Question 38

SL • Paper 2

Hard

Calculator Permitted

The function

f(x)=a\tan(b(x-c))+d

has vertical asymptotes at $x=1$ and $x=5$ . The centre of one branch is the point $(3,2)$ , and the point $(4,5)$ lies on the graph. Use the convention that $b>0$ and that $c$ is the $x$ -coordinate of the centre of the branch between the two given asymptotes.

Tangent graph with asymptotes at x=1 and x=5.

A

I.

Find the values of $b$ , $c$ and $d$ .

[3]

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II.

Find the value of $a$ .

[3]

Write your answer here...

B

Solve $f(x)=0$ for $-1\leq x\leq 9$ . Give your answers to 3 significant figures. If you did not obtain $f$ , use $f(x)=3\tan\left(\dfrac{\pi}{4}(x-3)\right)+2$ .

[3]

Write your answer here...

C

State the equations of all vertical asymptotes of $f$ in the interval $-1\leq x\leq 9$ .

[1]

Write your answer here...

0

Question 39

HL • Paper 2

Hard

Calculator Permitted

Consider the function

f(x)=3\sec(2x)-2,\quad 0\leq x\leq 2\pi

Separate continuous branches of y=3sec(2x)-2 on 0≤x≤2π, with gaps at the vertical asymptotes.

A

I.

Find the equations of the vertical asymptotes of the graph of $f$ .

[2]

Write your answer here...

II.

State the range of $f$ .

[3]

Write your answer here...

B

Solve $f(x)=4$ for $0\leq x\leq 2\pi$ .

[3]

Write your answer here...

C

Find the area enclosed by the curve $y=f(x)$ , the $x$ -axis, the $y$ -axis and the line $x=\dfrac{\pi}{6}$ .

[2]

Write your answer here...

0

Question 40

HL • Paper 2

Hard

Calculator Permitted

Consider the function

f(x)=5\sin x+12\cos x

where $0\leq x\leq 2\pi$ .

A

I.

Write $f(x)$ in the form $R\sin(x+\alpha)$ , where $R>0$ and $0<\alpha<\dfrac{\pi}{2}$ .

[4]

Write your answer here...

II.

State the maximum value of $f(x)$ .

[1]

Write your answer here...

B

Solve $f(x)=6$ for $0\leq x\leq 2\pi$ . If you did not obtain part (a), use $f(x)=13\sin(x+1.18)$ .

[3]

Write your answer here...

C

Let $A$ be the area between the graph of $y=f(x)$ and the $x$ -axis over the interval from the smaller solution in part (b) to the larger solution in part (b). Determine $A$ .

[4]

Write your answer here...

0

Question 41

HL • Paper 2

Hard

Calculator Permitted

Define

f(x)=\sin(\pi-x)+\cos(-x),\quad -\pi\leq x\leq 3\pi

A

I.

Use symmetry properties of trigonometric functions to simplify $f(x)$ .

[2]

Write your answer here...

II.

Show that, for any $x$ such that both $x$ and $\dfrac{\pi}{2}-x$ lie in the stated domain, $f\left(\dfrac{\pi}{2}-x\right)=f(x)$ . Hence the formula has reflection symmetry about the line $x=\dfrac{\pi}{4}$ on the portions of the graph for which reflected points are in the domain.

[2]

Write your answer here...

B

Write $f(x)$ in the form $R\sin(x+\alpha)$ , where $R>0$ and $0<\alpha<\dfrac{\pi}{2}$ .

[2]

Write your answer here...

C

Solve $f(x)=0.500$ for $-\pi\leq x\leq 3\pi$ .

[4]

Write your answer here...

0

Question 42

HL • Paper 3

Hard

Calculator Permitted

This question investigates the graph of the reciprocal trigonometric function $f_{a,d}(x)=a\sec\left(x-\frac{\pi}{6}\right)+d$ , where $a>0$ , for $0\le x\le 2\pi$ .

Transformed secant graph on $0\le x\le 2\pi$.

A

Consider first the function $f(x)=2\sec\left(x-\frac{\pi}{6}\right)-1$ .

I.

Find the equations of the vertical asymptotes of the graph of $f$ in the interval $0\le x\le 2\pi$ .

[2]

Write your answer here...

II.

Write down the range of $f$ .

[2]

Write your answer here...

B

Solve $2\sec\left(x-\frac{\pi}{6}\right)-1=3$ for $0\le x\le 2\pi$ .

[3]

Write your answer here...

C

For the general function $f_{a,d}$ , show that its range is $y\le d-a$ or $y\ge d+a$ .

[2]

Write your answer here...

D

member of the family has range $y\le -4$ or $y\ge 2$ . Determine $a$ and $d$ .

[2]

Write your answer here...

0

Question 43

HL • Paper 3

Hard

Calculator Permitted

For $0<a<\frac{\pi}{2}$ , define $g_a(x)=\sin(x+a)+\sin(x-a)$ for $0\le x\le 2\pi$ .

Three sample curves of g_a(x)=2cos(a)sin x.

A

Use compound angle identities to rewrite $g_a(x)$ .

I.

Show that $g_a(x)=2\cos a\sin x$ .

[3]

Write your answer here...

II.

State the amplitude and period of $g_a$ .

[2]

Write your answer here...

B

For $a=\frac{\pi}{6}$ , solve $g_a(x)=1$ for $0\le x\le 2\pi$ .

[3]

Write your answer here...

C

Determine the value of $a$ for which the equation $g_a(x)=1$ has exactly one solution in the interval $0\le x\le 2\pi$ .

[3]

Write your answer here...

0

Question 44

HL • Paper 3

Hard

Calculator Permitted

In triangle $ABC$ , side lengths $a$ , $b$ and $c$ are opposite angles $A$ , $B$ and $C$ respectively. Let $A=40^\circ$ and $a=8\ \text{cm}$ . The side $b$ may vary.

A labelled ambiguous-case triangle diagram with the point $A$, point $C$, and ray $AC$ fixed so that the angle at $A$ is $40^\circ$. Show the two possible points $B$ and $B'$ on the ray from $A$ for the same $b=AC$, with $BC=B'C=a=8\ \text{cm}$, and label the different lengths $c=AB$ and $c'=AB'$.

A

When $b=10\ \text{cm}$ , find the two possible values of angle $B$ .

[3]

Write your answer here...

B

For $b=10\ \text{cm}$ , find the two possible values of $c$ .

[3]

Write your answer here...

C

Determine the values of $b$ for which two different triangles are possible.

[3]

Write your answer here...

D

Describe what happens geometrically when $b=\frac{8}{\sin40^\circ}$ .

[2]

Write your answer here...

0

Question 45

HL • Paper 3

Hard

Calculator Permitted

The temperature $T$ degrees Celsius in a greenhouse during one day is modelled by $T(t)=d+a\cos(b(t-c))$ , where $0\le t\le 24$ is the time in hours after midnight and $a>0$ , $b>0$ , and $0\le c<24$ . The maximum temperature is $28^\circ\text{C}$ at $t=15$ , and the minimum temperature is $16^\circ\text{C}$ at $t=3$ . The model has period $24$ hours.

A

Determine three parameters in the model.

I.

Find $a$ and $d$ .

[3]

Write your answer here...

II.

Find $b$ .

[2]

Write your answer here...

B

Find $c$ for this model.

[3]

Write your answer here...

C

Find the first time after midnight when the model predicts $25^\circ\text{C}$ .

[2]

Write your answer here...

D

Find the rate of change of temperature at the time found in part (c), and interpret its sign.

[2]

Write your answer here...

0

Question 46

HL • Paper 1

Hard

Non Calculator

This question uses compound angle identities. Let

F(x)=\sqrt3\cos x-\sin x

A

I.

Show that $F(x)=2\cos\left(x+\dfrac{\pi}{6}\right)$ .

[3]

Write your answer here...

II.

Find the maximum value of $F(x)$ .

[2]

Write your answer here...

B

Solve $F(x)=1$ for $0\leq x\leq2\pi$ .

[4]

Write your answer here...

C

I.

Find the exact value of $F\left(\dfrac{7\pi}{12}\right)$ .

[2]

Write your answer here...

II.

Determine the values of $k$ for which $F(x)=k$ has no solution. Give your answer in interval notation.

[2]

Write your answer here...

0

Question 47

HL • Paper 1

Hard

Non Calculator

Let $0<x<\dfrac{\pi}{2}$ and let $t=\tan x$ .

A

I.

Use the compound angle formula for tangent to derive $\tan2x=\dfrac{2t}{1-t^2}$ , where $t\ne1$ .

[3]

Write your answer here...

II.

Explain why $t>0$ .

[2]

Write your answer here...

B

Solve $\tan2x=3\tan x$ for $0<x<\dfrac{\pi}{2}$ , excluding values for which $\tan2x$ is undefined.

[4]

Write your answer here...

C

I.

For $x=\dfrac{\pi}{6}$ , find the exact value of $\tan\left(2x+\dfrac{\pi}{4}\right)$ .

[2]

Write your answer here...

II.

Justify why $x=\dfrac{\pi}{4}$ is not a solution to the equation in part (b).

[2]

Write your answer here...

0

Question 48

HL • Paper 1

Hard

Non Calculator

For $0\leq x\leq2\pi$ , define

f(x)=\sin(\pi+x)+\sin(\pi-x)-2\cos(-x)

A

I.

Use symmetry properties to simplify $f(x)$ .

[3]

Write your answer here...

II.

State the range of $f$ .

[1]

Write your answer here...

B

Solve $f(x)=1$ for $0\leq x\leq2\pi$ .

[3]

Write your answer here...

C

I.

Prove that $f(2\pi-x)=f(x)$ for all $x$ in the domain.

[3]

Write your answer here...

II.

Interpret this result as a symmetry of the graph of $y=f(x)$ on $0\leq x\leq2\pi$ .

[2]

Write your answer here...

0

Question 49

HL • Paper 1

Hard

Non Calculator

Consider the function

g(x)=2\sec\left(x-\frac{\pi}{3}\right)-1

for $0\leq x\leq2\pi$ .

A

I.

Find the equations of the vertical asymptotes of the graph of $y=g(x)$ in the given interval.

[3]

Write your answer here...

II.

State the range of $g$ on its domain.

[2]

Write your answer here...

B

Solve $g(x)=3$ for $0\leq x\leq2\pi$ .

[4]

Write your answer here...

C

I.

Find the exact value of $g\left(\dfrac{\pi}{3}\right)$ .

[1]

Write your answer here...

II.

On the interval $\dfrac{\pi}{3}\leq x<\dfrac{5\pi}{6}$ , define the inverse branch $g^{-1}$ . Find $g^{-1}(5)$ .

[3]

Write your answer here...

0

Question 50

HL • Paper 2

Hard

Calculator Permitted

Let

g(x)=\arctan x+\arctan(1-x),\quad x\in\mathbb{R}

Graph of g(x)=arctan x+arctan(1-x) on a wide interval.

A

I.

Show that

\tan(g(x))=\frac{1}{x^2-x+1}

[3]

Write your answer here...

II.

Explain why $g(x)>0$ for all real $x$ .

[1]

Write your answer here...

B

Solve $g(x)=\dfrac{\pi}{4}$ .

[3]

Write your answer here...

C

Use a GDC to determine the maximum value of $g(x)$ and the value of $x$ at which it occurs.

[2]

Write your answer here...

D

Find the two solutions of $g(x)=0.800$ .

[2]

Write your answer here...

0

Question 51

HL • Paper 2

Hard

Calculator Permitted

Let

F(x)=\tan(2x)-3\tan x

where $0\leq x<2\pi$ and both terms are defined.

Graph of y = tan(2x) - 3 tan x on 0≤x<2π.

A

I.

Show that, for $u=\tan x$ ,

F(x)=\frac{u(3u^2-1)}{1-u^2}

[4]

Write your answer here...

B

Hence solve $F(x)=0$ for $0\leq x<2\pi$ .

[4]

Write your answer here...

C

Find all values of $x$ in $0\leq x<2\pi$ for which $F(x)=4$ .

[3]

Write your answer here...

0

Question 52

HL • Paper 2

Hard

Calculator Permitted

An alternating voltage is modelled by

V(t)=8\sin(50t)-6\cos(50t),\quad 0\leq t\leq0.200

where $t$ is measured in seconds and $V$ is measured in volts.

Sinusoidal voltage graph over 0 to 0.200 s with V=7 line.

A

I.

Show that $V(t)$ can be written in the form

V(t)=10\sin(50t-\alpha)

where $0<\alpha<\dfrac{\pi}{2}$ .

[4]

Write your answer here...

II.

Find the period of the voltage. Give your answer in seconds.

[1]

Write your answer here...

B

Find all times in $0\leq t\leq0.200$ for which $V(t)=7$ . If you did not obtain the form in part (a), use $V(t)=10\sin(50t-0.644)$ .

[4]

Write your answer here...

C

Determine the total length of time during $0\leq t\leq0.200$ for which $V(t)>7$ .

[3]

Write your answer here...

D

Find the first time after $t=0$ at which the voltage is increasing at its greatest possible rate.

[1]

Write your answer here...

0

Question 53

HL • Paper 3

Hard

Calculator Permitted

For $k>0$ , define $F_k(x)=\arctan x+\arctan(kx)$ for $x\ge 0$ . This question investigates equations of the form $F_k(x)=\alpha$ .

Family of curves F_k(x)=arctan x + arctan(kx) for selected positive k, with asymptote y=π.

A

Consider $F_k$ for a fixed positive value of $k$ .

I.

State the range of $F_k$ .

[2]

Write your answer here...

II.

Show that $F_k$ is increasing on $x\ge 0$ .

[2]

Write your answer here...

B

Solve $F_3(x)=\frac{\pi}{3}$ , giving your answer exactly.

[4]

Write your answer here...

C

Justify that for every $\alpha$ such that $0<\alpha<\pi$ , the equation $F_k(x)=\alpha$ has exactly one solution for $x\ge 0$ .

[2]

Write your answer here...

D

For $0<\alpha<\frac{\pi}{2}$ , let $t=\tan\alpha$ . Find the solution of $F_k(x)=\alpha$ in terms of $k$ and $t$ .

[2]

Write your answer here...

0

Question 54

HL • Paper 3

Hard

Calculator Permitted

A line through the origin makes an angle $\theta$ with the positive $x$ -axis, where $\tan\theta=\frac{1}{4}$ . A sequence of gradients is defined by $m_0=\tan\theta$ and $m_{n+1}=\frac{2m_n}{1-m_n^2}$ when $m_n^2\ne 1$ .

A coordinate diagram showing several lines through the origin whose angles are successive doublings of an initial acute angle, with the corresponding gradients labelled m_0, m_1 and m_2.

A

Find the first two gradients in the sequence.

I.

Find $m_1$ exactly.

[2]

Write your answer here...

II.

Find $m_2$ exactly.

[2]

Write your answer here...

B

Show by induction that $m_n=\tan(2^n\theta)$ whenever all the terms involved are defined.

[3]

Write your answer here...

C

Using technology, determine the least positive integer $n$ for which $m_n>10$ .

[2]

Write your answer here...

D

Interpret the result in part (c) geometrically.

[2]

Write your answer here...

0

Question 55

HL • Paper 3

Hard

Calculator Permitted

For real $k$ , consider the equation $2\sin x=k+\cos 2x$ on the interval $0\le x\le 2\pi$ .

Four sample curves comparing 2sin x with k + cos 2x.

A

Show that the equation can be written as $2\sin^2x+2\sin x+k-1=0$ .

[2]

Write your answer here...

B

Solve the equation for $k=0$ .

[3]

Write your answer here...

C

Determine the set of values of $k$ for which the equation has at least one solution.

[3]

Write your answer here...

D

Determine the value of $k$ for which the equation has exactly three solutions in the interval $0\le x\le 2\pi$ .

[3]

Write your answer here...

0

Question 56

HL • Paper 3

Hard

Calculator Permitted

For $0<r\le 1$ , define $f_r(x)=\arcsin x+\arccos(rx)$ . This question investigates how the chosen ranges of inverse trigonometric functions affect equations involving $f_r$ .

Selected curves of f_r(x)=arcsin x+arccos(rx) on [-1,1].

A

State the ranges of the two inverse functions used in $f_r$ .

[2]

Write your answer here...

B

Show that $f_1(x)=\frac{\pi}{2}$ for $-1\le x\le 1$ .

[2]

Write your answer here...

C

Solve $f_{\frac{1}{2}}(x)=\frac{2\pi}{3}$ exactly.

[4]

Write your answer here...

D

Find the value of $r$ for which $f_r(1)=\frac{2\pi}{3}$ .

[2]

Write your answer here...

0

Question 57

HL • Paper 3

Hard

Calculator Permitted

This question investigates the equation $4\cos^3x-3\cos x=\frac{1}{2}$ using compound angle identities.

Graphs of $y=\cos(3x)$ and $y=\frac{1}{2}$ on $0\le x\le 2\pi$, with the six intersection points marked. The horizontal line is at the positive level $y=0.5$; its y-axis tick label is $0.5$, not $-0.5$.

A

Derive a cubic expression for $\cos 3x$ in terms of $\cos x$ .

[4]

Write your answer here...

B

Hence solve $4\cos^3x-3\cos x=\frac{1}{2}$ for $0\le x\le 2\pi$ .

[4]

Write your answer here...

C

Deduce the number of distinct real roots of $8t^3-6t-1=0$ .

[2]

Write your answer here...

D

Find the three roots of $8t^3-6t-1=0$ to $3$ significant figures.

[2]

Write your answer here...

0

Question 58

HL • Paper 3

Hard

Calculator Permitted

Let $f(x)=\tan x+\tan\left(\frac{\pi}{3}-x\right)$ on the interval $-\frac{\pi}{6}<x<\frac{\pi}{2}$ , excluding any value where the function is undefined.

Curve of f(x)=tan x+tan(pi/3-x) on its domain.

A

Show that the graph of $f$ is symmetric about the vertical line $x=\frac{\pi}{6}$ .

[1]

Write your answer here...

B

Let $t=\tan x$ . Show that $f(x)=\frac{\sqrt{3}(1+t^2)}{1+\sqrt{3}t}$ .

[3]

Write your answer here...

C

Find the value of $x$ in $0\le x\le \frac{\pi}{3}$ for which $f(x)$ is a minimum, and find this minimum value.

[3]

Write your answer here...

D

Explain why $f(x)$ is undefined at $x=-\frac{\pi}{6}$ .

[2]

Write your answer here...

E

Solve $f(x)=4$ on $-\frac{\pi}{6}<x<\frac{\pi}{2}$ .

[3]

Write your answer here...

0

Question 59

HL • Paper 3

Hard

Calculator Permitted

A simplified sound wave is modelled by $S(t)=\sin(5t)+\sin(7t)$ for $0\le t\le \pi$ . This question investigates the zeros of this type of combined wave.

A

Use compound angle identities to show that $S(t)=2\sin(6t)\cos t$ .

[3]

Write your answer here...

B

Find all zeros of $S(t)$ in the interval $0\le t\le \pi$ .

[3]

Write your answer here...

C

For an integer $n\ge 2$ , define $S_n(t)=\sin((n-1)t)+\sin((n+1)t)$ . Show that $S_n(t)=2\sin(nt)\cos t$ .

[3]

Write your answer here...

D

Deduce the number of distinct zeros of $S_n(t)$ in $0\le t\le \pi$ .

[3]

Write your answer here...

0

Question 60

HL • Paper 3

Hard

Calculator Permitted

Consider $f(x)=\sec x+\csc x$ on the interval $0<x<\frac{\pi}{2}$ .

Graph of y = sec x + csc x on 0 < x < π/2, showing a single minimum and steep growth near both ends.

A

Rewrite $f(x)$ in terms of $u=\sin x+\cos x$ .

I.

Show that $f(x)=\frac{\sin x+\cos x}{\sin x\cos x}$ .

[1]

Write your answer here...

II.

Let $u=\sin x+\cos x$ . Hence show that $f(x)=\frac{2u}{u^2-1}$ .

[3]

Write your answer here...

B

Find the range of possible values of $u$ on $0<x<\frac{\pi}{2}$ .

[2]

Write your answer here...

C

Determine the minimum value of $f(x)$ and the value of $x$ where it occurs.

[4]

Write your answer here...

D

Solve $\sec x+\csc x=4$ for $0<x<\frac{\pi}{2}$ .

[3]

Write your answer here...

0