Trigonometry, grade 10 (5 units)

Sine, cosine and tangent are functions that repeat themselves: every full turn of the unit circle brings them back to the same value, so they describe any periodic process – a Ferris wheel, the tides, the length of the day – and they also work out the sides and angles of any triangle.Subjects

The topics, in order:

the trigonometric functions on the unit circle (in radians): winding a number onto the unit circle, degrees and radians, the length of an arc and the area of a sector, sine and cosine, special values, symmetry and period;
the graphs: sine and cosine, shifts, stretches and squeezes, periodic processes, the tangent function;
trigonometric equations: basic equations, the general solution and the solutions in an interval, a linear argument, equations solved with identities;
trigonometry in the plane (in degrees): the right-angled triangle, the area of a triangle by the sine, the law of sines, the law of cosines, and solving problems in geometry.

Each lesson says whether its angles are measured in radians or in degrees.

Two kinds of lessons:

learning lessons – an explanation, solved examples and a few questions to try;
practice lessons – questions only.

How the page works: Using a page.

Last updated on 18.10.2026

Formulas: trigonometry, grade 10 (5 units)

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The unit circle (radians)

The unit circle
$\displaystyle x^2+y^2=1$
The point of the number $\displaystyle t$ on the unit circle
$\displaystyle P(t)=(\cos t,\ \sin t)$
All the numbers with the same point as $\displaystyle t$ ( $\displaystyle k$ an integer)
$\displaystyle t+2\pi k$
Radians and degrees
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle \ell$ $\displaystyle =$ $\displaystyle \alpha R$
The area of a sector of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha R^2$

Special values

$\displaystyle \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}$
$\displaystyle \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$
$\displaystyle \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$
$\displaystyle \tan\frac{\pi}{6}=\frac{\sqrt{3}}{3},\quad\tan\frac{\pi}{4}=1,\quad\tan\frac{\pi}{3}=\sqrt{3}$

Identities

The basic identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^2\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \sin(\alpha+2\pi k)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \tan(\alpha+\pi k)$ $\displaystyle =$ $\displaystyle \tan\alpha$

Graphs of trigonometric functions

A general sine (and a cosine likewise)
$\displaystyle y=A\sin(ax+b)+B$
The period of $\displaystyle \sin(ax+b)$ and of $\displaystyle \cos(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{2\pi}{|a|}$
The range of $\displaystyle A\sin(ax+b)+B$
$\displaystyle B-|A|\leq y\leq B+|A|$
The period of $\displaystyle \tan(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\pi}{|a|}$

Trigonometric equations ($k$ an integer)

The solutions of $\displaystyle {\sin x}={\sin\alpha}$
$\displaystyle x=\alpha+2\pi k,\quad x=\pi-\alpha+2\pi k$
The solutions of $\displaystyle {\cos x}={\cos\alpha}$
$\displaystyle x=\pm\alpha+2\pi k$
The solutions of $\displaystyle {\tan x}={\tan\alpha}$
$\displaystyle x=\alpha+\pi k$

Trigonometry in a triangle (degrees)

In a right-angled triangle: $\displaystyle a$ the leg opposite $\displaystyle \alpha$ , $\displaystyle b$ the leg beside it, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha=\frac{a}{c},\quad\cos\alpha=\frac{b}{c},\quad\tan\alpha=\frac{a}{b}$
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
The area of a triangle ( $\displaystyle \alpha$ – the angle between the sides $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}bc\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ and $\displaystyle \gamma$ – the angles at the side $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^2\sin\beta\sin\gamma}{2\sin(\beta+\gamma)}$
The law of sines ( $\displaystyle R$ – the radius of the circle that circumscribes the triangle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between the sides $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$