Sine and cosine on the unit circle

A unit circle is a circle with centre at the origin and radius $1$. Suppose the terminal arm of an angle $\theta$ meets the circle at $P$. The coordinates of $P$ are $(\cos \theta,\sin \theta)$, where $\theta$ is measured anticlockwise from the positive $x$-axis, usually in radians unless degrees are explicitly given.

Here, $\cos \theta$ gives the $x$-coordinate of $P$, while $\sin \theta$ gives its $y$-coordinate. Unlike right-triangle SOHCAHTOA, this definition works for any angle, including angles larger than $\frac{\pi}{2}$ and negative angles.

The quadrant determines each sign. Both coordinates are positive in quadrant I. In quadrant II, $x$ is negative and $y$ is positive, so cosine is negative and sine is positive. Both are negative in quadrant III. In quadrant IV, cosine is positive and sine is negative. This explains the usual quadrant sign diagram.

Related positions on the unit circle produce related trig values. For example,

Reflecting a point in the $x$-axis doesn’t change its $x$-coordinate. Also,

since a rotation by $\pi$ moves a point to the opposite side of the circle. And

because $3\pi-x$ has the same tangent behaviour as $\pi-x$. Tangent is negative in quadrant II when the reference angle is $x$.

Tangent and the gradient of a line

The tangent function is defined by

where $\theta$ is the angle and $\cos \theta \ne 0$. Since $\sin \theta$ is the vertical coordinate and $\cos \theta$ is the horizontal coordinate, tangent compares vertical change with horizontal change.

For a straight line through the origin that makes an angle $\theta$ with the positive $x$-axis, the equation is

where $x$ is the horizontal coordinate and $y$ is the vertical coordinate. So $\tan \theta$ is the gradient of the line. This connects trigonometry with coordinate geometry: angle and gradient are two ways to represent the same steepness.

Exact values

You need to know the exact values of sine, cosine and tangent for $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, and their multiples. The first-quadrant values come from two special triangles: the $30^\circ$-$60^\circ$-$90^\circ$ triangle and the $45^\circ$-$45^\circ$-$90^\circ$ triangle.

For the first quadrant:

• $\sin 0=0$, $\sin \frac{\pi}{6}=\frac{1}{2}$, $\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}$, $\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}$, $\sin \frac{\pi}{2}=1$.
• Cosine uses the same list in reverse order.
• Tangent is sine divided by cosine, giving $\tan \frac{\pi}{6}=\frac{1}{\sqrt{3}}$, $\tan \frac{\pi}{4}=1$, and $\tan \frac{\pi}{3}=\sqrt{3}$.

For multiples, don’t memorise a huge table. Find the reference angle first, then use the quadrant to decide the sign. For example, $\cos \frac{3\pi}{4}=-\frac{1}{\sqrt{2}}$ because $\frac{3\pi}{4}$ lies in quadrant II and has reference angle $\frac{\pi}{4}$. Similarly, $210^\circ$ lies in quadrant III with reference angle $30^\circ$, so $\tan 210^\circ=\frac{\sqrt{3}}{3}$. Exact trig values with reference-angle examples

The ambiguous case of the sine rule

The ambiguous case occurs in a non-right triangle when the sine rule gives two possible triangles, since two supplementary angles have the same sine. In symbols,

for angles inside a triangle.

The sine rule is

where $A$, $B$ and $C$ are the angles of a triangle, and $a$, $b$ and $c$ are their opposite side lengths, all measured in the same length unit. The ambiguous case can arise when two sides and a non-included acute angle are given, especially if the side opposite the given angle is the shorter of the two given sides.

Use this routine: calculate the angle with the sine rule, check its supplement, then see whether the triangle angle sum allows that second value. The largest angle must also be opposite the longest side. This check rules out many impossible second answers.

Historically, sine as a function of an angle appears explicitly in Indian mathematics, including work associated with Aryabhata. Trigonometry did not arrive fully formed from one place; different cultures shaped the representation and language now treated as standard mathematics.

The Pythagorean identity

A trigonometric identity is an equation involving trigonometric functions that holds for every angle where both sides are defined. The most important identity in this part of the course is

It comes directly from Pythagoras on the unit circle. Since the point $(\cos \theta,\sin \theta)$ is one unit from the origin, its coordinates must satisfy $x^2+y^2=1$.

This identity lets you move between sine and cosine without calculating the angle. For example, if $\sin \theta=\frac{3}{5}$, then

so $\cos \theta=\pm\frac{4}{5}$. The sign matters and depends on the quadrant. Cosine is negative when the angle lies in quadrant II, while it is positive in quadrant I.

Double angle identities for sine and cosine

The double angle identities express a trigonometric function of twice an angle using functions of the original angle. You should know

and

By using $\cos^2 \theta+\sin^2 \theta=1$, you can also write the cosine double angle identity as

or

All three versions of $\cos 2\theta$ are equivalent. Usually, though, one form fits the question better than the others. For an equation containing only sine, choose $1-2\sin^2 \theta$. For one containing only cosine, use $2\cos^2 \theta-1$.

Dynamic graphing packages help you check these identities visually. Compare $\sin 2x$ with $2\sin x\cos x$, or plot the three forms of $\cos 2x$. Their graphs lie exactly on top of each other. This gives visual evidence of the identity rather than a proof by itself, but it’s a useful way to build confidence in the algebra.

Relationships between trigonometric ratios

The basic relationship

where $\cos \theta \ne 0$, allows you to find possible tangent values from information about sine and cosine. Use it alongside the Pythagorean identity and the signs for each quadrant.

If $\sin \theta$ is known but $\theta$ is not, begin by finding the possible values of $\cos \theta$ from $\cos^2 \theta=1-\sin^2 \theta$. Then divide. When no quadrant is given, tangent may have more than one possible value. If the angle is stated to be acute, sine and cosine are both positive, so the ambiguity disappears.

A common neat application is finding $\sin 2x$ without finding $x$. If $\cos x=\frac{3}{4}$ and $x$ is acute, then

and therefore

Pay attention to the phrase without finding $x$. It isn’t a trick; it is the purpose of using identities. They change the representation without bringing in unnecessary rounded angle values.

Circular functions and their graphs

A circular function is a trigonometric function with an input interpreted as an angle on the unit circle. This section covers $\sin x$, $\cos x$ and $\tan x$. Here, $x$ is the input angle and is measured in radians unless the question specifies degrees.

A periodic function repeats its output values after a fixed horizontal interval. Both sine and cosine have period $2\pi$, so

and

The period of tangent is $\pi$. Therefore,

Over one full period, the sine graph starts at $0$, rises to $1$, returns to $0$, falls to $-1$, and comes back to $0$. Cosine starts at $1$ and has the same wave shape, shifted left by $\frac{\pi}{2}$. Tangent looks different. It increases through the origin, with vertical asymptotes at values such as $x=-\frac{\pi}{2}$, $x=\frac{\pi}{2}$ and $x=\frac{3\pi}{2}$.

Amplitude, period and composite sine functions

A transformed sine function is usually written as

Here, $x$ is the input angle or time-like variable, while $f(x)$ is the output value. The value $a$ sets the vertical scale, $b$ is the horizontal frequency factor, $c$ controls horizontal translation, and $d$ gives the vertical translation. These quantities are unitless in pure trigonometry. In a modelling context, $x$ and $f(x)$ use the units stated in the problem.

The amplitude measures the distance from the midline to either a maximum or a minimum. For $f(x)=a\sin(b(x+c))+d$, it is $|a|$. When the maximum and minimum values are known, use

The period is the horizontal length of one complete cycle. For this function,

Its midline is $y=d$. The maximum and minimum values give

Watch the sign of $c$. In $\sin(b(x+c))$, a positive $c$ shifts the graph left by $c$. Many textbooks instead use $a\sin(b(x-c))+d$; in that form, positive $c$ shifts the graph right. Read the brackets rather than relying on memory.

Transformations

Trigonometric graphs follow the same transformation rules as other functions. Moving from $y=\sin x$ to $y=3\sin 2x$, the outside factor $3$ produces a stretch by scale factor $3$ in the $y$ direction. The factor $2$ inside the angle produces a stretch by scale factor $\frac{1}{2}$ in the $x$ direction. The transformed graph has amplitude $3$ and period $\pi$.

Consider the function

The negative sign reflects the sine curve in the $x$-axis. Inside the brackets, $+\frac{\pi}{2}$ shifts it left by $\frac{\pi}{2}$, while the outside $+1$ shifts it up by $1$. These are the same graph transformations used for functions.

Real-life contexts

Repeated motion can often be modelled with trigonometric functions. Examples include tide height, a point on a Ferris wheel, simple harmonic motion, and periodic sound waves in music. Each parameter has a clear meaning: amplitude is half the total variation, period is the length of one full cycle, and vertical shift gives the central value.

Suppose a tide ranges from $7$ metres to $15$ metres. Its amplitude is $4$ metres and its midline is $11$ metres. If one cycle takes $12$ hours, then $\frac{2\pi}{|b|}=12$, giving $b=\frac{\pi}{6}$ when time is measured in hours. One possible cosine model is

where $t$ is the time in hours after a chosen starting time, and $h(t)$ is the water depth in metres. Units matter here: $\frac{\pi}{6}$ means radians per hour.

Regression technology may produce a sinusoidal model in an algebraic form different from $a\sin(b(x+c))+d$. The model isn’t wrong, but its parameters may need to be rewritten or interpreted carefully before you compare it with the syllabus form.

Why the interval matters

A trigonometric equation is an equation in which the unknown appears inside a trigonometric function. Trigonometric functions are periodic, so most trig equations have infinitely many solutions when no interval is given. In this syllabus, you solve them over a finite interval, such as $0\le x\le 2\pi$ or $0^\circ\le x\le 180^\circ$. General solutions are not required here.

Assume radians on examination papers unless the question clearly uses degrees. An interval in radians needs an answer in radians; an interval in degrees needs an answer in degrees. Don’t switch between them halfway through your solution.

Solving analytically

A clean analytic method is:

1. Isolate the trigonometric expression if possible.
2. Find the reference angle or principal value.
3. Use the unit circle or graph to find all solutions in the stated finite interval.
4. Check endpoints if the interval includes them.

For example, start with

which gives $\sin x=\frac{1}{2}$. On $0\le x\le 2\pi$, sine is positive in quadrants I and II. Therefore,

If the equation contains a multiple or shifted angle, solve for the complete inside expression first. For

set

Convert the given interval for $x$ to the corresponding interval for $u$, then solve $\tan u=\frac{1}{2}$ within that interval. Finally, convert back using $x=\frac{u}{3}+4$. Doing this helps you avoid missing repeated solutions.

Solving graphically

For a graphical solution, plot both sides and locate their intersections within the given interval. With

graph $y=2\sin x$ and $y=1$. The solutions are the $x$-coordinates of the intersection points. This method is useful when rearranging the equation is awkward, but the interval and angle mode still need to be correct.

Equations that become quadratics

Some trig equations can be rewritten as quadratic equations in $\sin x$, $\cos x$ or $\tan x$. Temporarily treat the trigonometric function as one variable.

For example,

is easier to read as a quadratic in $\sin x$:

So $\sin x=0$ or $\sin x=-\frac{1}{2}$. From there, find every value of $x$ in the finite interval.

A less obvious example is

Use $\cos 2x=1-2\sin^2 x$ to obtain

then rearrange it into a quadratic in $\sin x$. Reject impossible values such as $\sin x=\frac{3}{2}$, since sine and cosine values must lie between $-1$ and $1$.

When an equation contains both sine and cosine, look for an identity that rewrites everything using one trig function. This is where the double angle identities from section 3.6 earn their keep.

Reciprocal trigonometric ratios

A reciprocal trigonometric ratio is a trigonometric function formed by taking the reciprocal of sine, cosine or tangent. There are three such ratios:

and

Some resources use cosec instead of $\csc$; both mean the reciprocal of sine.

Their graphs have vertical asymptotes wherever the original function equals zero. For instance, $\sec x$ has vertical asymptotes where $\cos x=0$, while $\csc x$ has them where $\sin x=0$. Reciprocal trig ratios and asymptotes

Pythagorean identities with reciprocal ratios

Begin with

and divide each term by $\cos^2 \theta$. This gives

It is usually written as

Dividing the original identity by $\sin^2 \theta$ instead gives

These identities come from the same unit-circle relationship, expressed using reciprocal ratios. In proofs, they often provide the quickest way to replace $1+\tan^2 \theta$ or $1+\cot^2 \theta$ with one squared reciprocal function.

Inverse trigonometric functions

An inverse trigonometric function returns an angle from a trigonometric ratio after the original trigonometric function has been restricted to a one-to-one domain. The restriction is necessary because $\sin x$, $\cos x$ and $\tan x$ repeat, so without it they fail the horizontal line test.

The inverse sine function is

Its domain is $[-1,1]$, and its range is $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

The inverse cosine function is

Its domain is $[-1,1]$, and its range is $[0,\pi]$.

The inverse tangent function is

Its domain is $\mathbb{R}$, and its range is $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. The endpoints aren’t included because the graph has horizontal asymptotes there. Inverse trig branches, domains and ranges.

These ranges are conventional, but they aren’t arbitrary. Each one selects a branch on which the original trigonometric function is one-to-one. The chosen range also determines the value we state. For example, $\arcsin\left(\frac{1}{2}\right)$ is $\frac{\pi}{6}$ rather than $\frac{5\pi}{6}$ because $\frac{\pi}{6}$ lies in the selected range for arcsine.

Compound angle identities

A compound angle identity rewrites a trigonometric function of a sum or difference of angles using the trigonometric functions of the separate angles. For angles $A$ and $B$, the sine identities are

and

For cosine, the identities are

and

The tangent identities are

and

provided the denominators are not zero and the tangent values involved are defined.

You can use these identities to find exact values for angles formed from standard angles. For instance, $165^\circ$ can be written as $120^\circ+45^\circ$. Its tangent can therefore be found exactly with the compound tangent identity instead of using a calculator decimal.

Deriving double angle identities

A double angle identity is a compound angle identity in which both angles are the same.

This gives

Next

which gives

The other two forms of $\cos 2\theta$ follow from $\sin^2 \theta+\cos^2 \theta=1$.

Double angle identity for tangent

Set $A=B=\theta$ in the compound tangent identity to get

where $\tan \theta$ is defined and $1-\tan^2 \theta\ne 0$. This denominator condition matters. If it is zero, the angle $2\theta$ corresponds to a tangent asymptote.

Compound angle identities also provide some of the algebraic background for De Moivre's theorem, which appears later in complex numbers. The same ideas underlie phase shifts in sinusoidal voltage and the angle relationships used in triangulation, including GPS-style positioning.

Symmetry from the unit circle and the graphs

Trigonometric symmetry isn't a collection of unrelated rules. It follows from reflections on the unit circle and the repeating shapes of trigonometric graphs. For an angle $\theta$,

The two angles are mirror images in the $y$-axis, so their $y$-coordinates are equal.

Also,

Here, reflection gives the point the opposite $x$-coordinate.

Dividing sine by cosine gives

Quadrant II reference-angle identities on the unit circle.

These are the quadrant-II forms of the reference-angle relationships. They link back to the unit circle definitions in section 3.5 and the compound angle identities in section 3.10. For example, the compound identity for $\sin(A-B)$ can also be used to prove $\sin(\pi-\theta)$.

Odd and even symmetry

An even function satisfies $f(-x)=f(x)$ for every $x$ in its domain. Cosine is even:

An odd function satisfies $f(-x)=-f(x)$ for every $x$ in its domain. Sine and tangent are odd:

and

On a graph, even symmetry appears as reflection in the $y$-axis. Odd symmetry gives rotational symmetry of $180^\circ$ about the origin. So the cosine graph is balanced on either side of the $y$-axis, while sine and tangent pass through the origin and show matching opposite behaviour on each side.

Periodicity and repeated solutions

Symmetry makes equations easier to solve because one known solution, together with periodicity, often produces others. For example, if $\sin x=k$ has one solution in quadrant I, the identity $\sin(\pi-\theta)=\sin \theta$ may give another in quadrant II. Further solutions then repeat every $2\pi$.

A trigonometric equation can therefore have infinitely many discrete solutions on the real line but only finitely many within a finite interval. There’s no contradiction: periodicity produces endless repeats, yet each repeat is a separate point rather than a continuous block of solutions. Simple harmonic motion graphs in physics use exactly this repeated structure.