Sine, cosine and tangent of any angle

By the end of this lesson

By the end you can find $\sin\theta$ , $\cos\theta$ and $\tan\theta$ for any angle (obtuse, reflex, negative, bigger than $360^\circ$ ), say whether each is positive or negative, and rewrite it using an acute angle.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\theta$	"theta": the angle we want the sin, cos or tan of
$\alpha$	"alpha": the basic angle, an acute angle (between $0^\circ$ and $90^\circ$ ) linked to $\theta$
quadrant	one of the four quarters the axes split the plane into, numbered 1 to 4 anticlockwise from top right

SOH CAH TOA has a 90° problem

Right-angled triangles can't have an angle of $150^\circ$ . Yet your calculator happily tells you $\sin 150^\circ = 0.5$ . So what is it the sine of?

The fix: stop using triangles as the definition and use a circle instead.

Definition

Sine and cosine of any angle

The unit circle is the circle $x^2 + y^2 = 1$ : centre $(0, 0)$ , radius 1.

Start at the point $(1, 0)$ on it. Turn through the angle $\theta$ :

anticlockwise if $\theta$ is positive
clockwise if $\theta$ is negative

You land at a point $P$ . Then, by definition:

$P = (\cos\theta, \sin\theta)$

$\cos\theta$ is the $x$ -coordinate of $P$ .
$\sin\theta$ is the $y$ -coordinate of $P$ .
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ , as long as $\cos\theta \neq 0$ .

Why this agrees with triangles

For an acute angle, drop a line straight down from $P$ to the $x$ -axis. You get a right-angled triangle with hypotenuse $1$ (the radius).

"Opposite over hypotenuse" is $\dfrac{y}{1} = y$ : the height. That's $\sin\theta$ .
"Adjacent over hypotenuse" is $\dfrac{x}{1} = x$ : the width. That's $\cos\theta$ .

And $\tan\theta = \dfrac{y}{x}$ is opposite over adjacent: TOA again.

Same answers as GCSE. But now the definition keeps working past $90^\circ$ .

Turning through more than $360^\circ$ just goes round again. So $\sin 400^\circ = \sin 40^\circ$ , because $400^\circ = 360^\circ + 40^\circ$ .

Quick check

Fill in the blanksQuestion 1 · Recall

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