Trigonometric Functions - IB Math AA Question Bank

Subjects

Features

Review

HOT

Tutoring

Trigonometric Functions

Practice exam-style IB Math AA questions for Trigonometric Functions, aligned with the syllabus and grouped by topic.

Verified by Karim

Paper

Difficulty

Status

Level

This content can't be printed from the browser. Use Export PDF with Pro for an offline copy.

Question 1

SL • Paper 1

Easy

Non Calculator

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 $\tan\theta=\sqrt{3}$ :

Find the value of $\theta$ .

[2]

Write down the coordinates of $P$ .

[2]

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

[1]

Question 2

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

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

Find $\sin x$ .

[2]

Find $\cos 2x$ .

[1]

Find $\sin 2x$ .

[2]

Question 3

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

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

[4]

Question 4

SL • Paper 2

Easy

Calculator Permitted

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.

Find $\cos\theta$ .

[1]

Find the exact value of $k$ .

[2]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 5

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

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

Find $\sin x$ .

[2]

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

[3]

Question 6

SL • Paper 1

Medium

Non Calculator

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.

Find the possible values of angle $C$ .

[5]

Question 7

SL • Paper 1

Medium

Non Calculator

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 $t$ , $f(t)>f(0)$ .

Find the amplitude and the value of $d$ .

[2]

Find the value of $b$ .

[2]

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

[2]

Question 8

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

[5]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

Find $\sec\theta$ .

[3]

Find $\csc\theta$ .

[2]

Question 10

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

Find the domain of $f$ .

[2]

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

[3]

Question 11

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

Find $\tan 2\theta$ .

[3]

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

[2]

Question 12

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

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

[2]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 13

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Consider the equation

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

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

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

[1]

Hence solve the equation.

[3]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

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

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

[3]

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

[3]

Question 15

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

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

State the domain and range of $f$ .

[2]

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

[3]

Question 16

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

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

Find the equations of the vertical asymptotes in this interval.

[2]

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

[2]

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

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

This question uses the compound angle identities.

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

[3]

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

[3]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

[6]

Question 19

SL • Paper 2

Medium

Calculator Permitted

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.

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

[3]

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

[3]

Question 20

SL • Paper 2

Medium

Calculator Permitted

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 $-6<c\le 6$ .

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

[3]

Find the value of $c$ .

[2]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 21

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The equation

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

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

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

[1]

Hence solve the equation for $x$ .

[4]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

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

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

[2]

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

[1]

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

[3]

Question 23

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

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

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

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

[2]

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

[2]

Solve $f(x)=1$ .

[2]

Question 24

SL • Paper 1

Medium

Non Calculator

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 $\theta$ .

I.

Find $\sin\theta$ .

[2]

II.

Hence find $\tan\theta$ .

[2]

I.

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

[2]

II.

Write down an equation of $l$ .

[1]

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]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 25

SL • Paper 1

Medium

Non Calculator

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 $b=2\sqrt{6}$ .

I.

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

[2]

II.

Find the two possible values of $B$ .

[2]

I.

Find the corresponding two possible values of $C$ .

[2]

II.

Find the two possible exact values of $c$ .

[2]

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

[3]

Question 26

SL • Paper 1

Medium

Non Calculator

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

I.

State the amplitude and period of $f$ .

[2]

II.

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

[2]

I.

Find the maximum and minimum values of $f$ .

[2]

II.

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

[2]

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

[3]

Question 27

SL • Paper 2

Medium

Calculator Permitted

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.

I.

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

[2]

II.

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

[2]

III.

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

[2]

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]

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

[3]

Question 28

SL • Paper 2

Medium

Calculator Permitted

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 $b=13.0\ \text{cm}$ .

I.

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

[4]

I.

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

[2]

II.

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

[2]

A 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]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 29

SL • Paper 2

Medium

Calculator Permitted

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 $(7,1)$ .

I.

Find the amplitude and the value of $d$ .

[2]

II.

Find the value of $b$ .

[2]

III.

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

[3]

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

[2]

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]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

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

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

[2]

Hence find the exact solution of the original equation.

[3]

Question 31

SL • Paper 1

Hard

Non Calculator

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.

I.

Find $a$ and $d$ .

[2]

II.

Find $b$ .

[2]

III.

Find a possible value of $c$ .

[1]

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

[2]

I.

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

[3]

II.

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

[2]

Question 32

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Consider the equation

2\sin x=\cos 2x

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

I.

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

[3]

II.

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

[2]

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]

I.

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

[2]

II.

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 33

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Consider the equation

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

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

I.

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

[2]

II.

Solve the equation for $u$ in this interval.

[2]

Hence solve the original equation for $x$ .

[3]

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]

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]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

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

\sec x+\tan x=3

AI.

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

[2]

AII.

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

[2]

BI.

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

[2]

BII.

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

[2]

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

[4]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The function $f$ is defined by

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

AI.

Find the domain of $f$ .

[2]

AII.

State the range of $f$ .

[2]

BI.

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

[2]

BII.

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

[2]

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]

Question 36

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Consider the function

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

I.

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

[2]

I.

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

[4]

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

[3]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 37

SL • Paper 2

Hard

Calculator Permitted

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 $(4,y)$ .

I.

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

[2]

II.

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

[2]

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

[3]

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

[2]

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]

Question 38

SL • Paper 2

Hard

Calculator Permitted

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.

I.

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

[3]

II.

Find the value of $a$ .

[3]

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]

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

[1]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the function

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

I.

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

[2]

II.

State the range of $f$ .

[3]

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

[3]

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

[2]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the function

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

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

I.

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

[4]

II.

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

[1]

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]

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]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Define

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

I.

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

[2]

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

[2]

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

[4]

Question 42

HL • Paper 3

Hard

Calculator Permitted

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

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]

II.

Write down the range of $f$ .

[2]

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

[3]

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

[2]

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

[2]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

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

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

I.

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

[3]

II.

State the amplitude and period of $g_a$ .

[2]

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

[3]

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]

Question 44

HL • Paper 3

Hard

Calculator Permitted

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.

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

[3]

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

[3]

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

[3]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 45

HL • Paper 3

Hard

Calculator Permitted

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.

Determine three parameters in the model.

I.

Find $a$ and $d$ .

[3]

II.

Find $b$ .

[2]

Find $c$ for this model.

[3]

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

[2]

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

[2]

Question 46

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

This question uses compound angle identities. Let

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

I.

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

[3]

II.

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

[2]

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

[4]

I.

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

[2]

II.

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

[2]

Question 47

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

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

I.

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

[3]

II.

Explain why $t>0$ .

[2]

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

[4]

I.

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

[2]

II.

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

[2]

Question 48

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

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

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

I.

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

[3]

II.

State the range of $f$ .

[1]

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

[3]

I.

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

[3]

II.

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 49

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the function

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

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

I.

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

[3]

II.

State the range of $g$ on its domain.

[2]

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

[4]

I.

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

[1]

II.

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

[3]

Question 50

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

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

I.

Show that

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

[3]

II.

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

[1]

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

[3]

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

[2]

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

[2]

Question 51

HL • Paper 2

Hard

Calculator Permitted

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.

I.

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

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

[4]

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

[4]

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

[3]

Question 52

HL • Paper 2

Hard

Calculator Permitted

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.

I.

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

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

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

[4]

II.

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

[1]

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]

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

[3]

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

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 53

HL • Paper 3

Hard

Calculator Permitted

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 $F_k(x)=\alpha$ .

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

I.

State the range of $F_k$ .

[2]

II.

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

[2]

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

[4]

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]

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]

Question 54

HL • Paper 3

Hard

Calculator Permitted

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 $m_n^2\ne 1$ .

Find the first two gradients in the sequence.

I.

Find $m_1$ exactly.

[2]

II.

Find $m_2$ exactly.

[2]

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

[3]

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

[2]

Interpret the result in part (c) geometrically.

[2]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

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

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

[2]

Solve the equation for $k=0$ .

[3]

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

[3]

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

[3]

Question 56

HL • Paper 3

Hard

Calculator Permitted

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 $f_r$ .

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

[2]

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

[2]

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

[4]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

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

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

[4]

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

[4]

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

[2]

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

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

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.

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

[1]

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

[3]

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]

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

[2]

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

[3]

Question 59

HL • Paper 3

Hard

Calculator Permitted

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.

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

[3]

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

[3]

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]

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

[3]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

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

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]

II.

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

[3]

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

[2]

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

[4]

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

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Geometry & Shapes

Vectors

The Ultimate All-in-One
IB Platform

All content on this website has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate and IB are registered trademarks owned by the International Baccalaureate Organization.