Why the Sine of 150° Is Exactly the Same as the Sine of 30°

Right-triangle definitions only work for acute angles. The unit circle works for every angle. Draw a circle of radius $1$ centred at the origin, and rotate a ray through an angle $x$ from the positive x-axis. The ray meets the circle at a point $P$:

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

So cosine is the x-coordinate and sine is the y-coordinate, for any angle at all. This part covers the six functions, the fundamental identity, signs and allied angles, and graphs.

**If the ray at angle $x$ meets the unit circle at $(a,b)$, then $\cos x=a$ and $\sin x=b$, and the other four functions are $\tan x=\frac{b}{a}$, $\cot x=\frac{a}{b}$, $\sec x=\frac{1}{a}$ and $\mathrm{cosec}x=\frac{1}{b}$, wherever the denominator is not zero.

Values at quadrantal angles**, read from the points $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$:

$$\sin 0=0,\cos 0=1\sin \frac{\pi }{2}=1,\cos \frac{\pi }{2}=0$$
$$\sin \pi =0,\cos \pi =-1\sin \frac{3\pi }{2}=-1,\cos \frac{3\pi }{2}=0$$

Zeros. $\sin x=0$ when $x=n\pi$, and $\cos x=0$ when $x=(2n+1)\frac{\pi }{2}$, for any integer $n$.

Worked example. At $x=\frac{\pi }{3}$, the point is $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$, so

$$\tan \frac{\pi }{3}=\sqrt{3}\sec \frac{\pi }{3}=2\mathrm{cosec}\frac{\pi }{3}=\frac{2}{\sqrt{3}}$$

An everyday example. **The tip of a $1$ m ceiling fan blade** is at the point $(\cos x,\sin x)$ after turning through angle $x$.

The substance. **$\tan x$ and $\sec x$ are undefined wherever $\cos x=0$**, at odd multiples of $\frac{\pi }{2}$.

**The point $(\cos x,\sin x)$ lies on a circle of radius $1$, so by Pythagoras ${\cos }^{2}x+{\sin }^{2}x=1$; dividing by ${\cos }^{2}x$ or ${\sin }^{2}x$ gives $1+{\tan }^{2}x={\sec }^{2}x$ and $1+{\cot }^{2}x={\mathrm{cosec}}^{2}x$.

Worked example 1.** $\cos x=-\frac{3}{5}$ and $x$ is in quadrant III.

$${\sin }^{2}x=1-\frac{9}{25}=\frac{16}{25}\sin x=-\frac{4}{5}\text{(negative in QIII)}$$
$$\tan x=\frac{4}{3}\cot x=\frac{3}{4}\sec x=-\frac{5}{3}\mathrm{cosec}x=-\frac{5}{4}$$

Worked example 2. Simplify $(1-{\cos }^{2}x)(1+{\cot }^{2}x)$.

$${\sin }^{2}x\times {\mathrm{cosec}}^{2}x=1$$

An everyday example. **However far a fan blade turns, its tip stays exactly $1$ m from the centre** — which is the identity ${\cos }^{2}x+{\sin }^{2}x=1$ in action.

The substance. **The identity holds for every real $x$**, including obtuse, reflex and negative angles, not just acute ones.

All functions are positive in quadrant I, only sine and cosecant in quadrant II, only tangent and cotangent in quadrant III, and only cosine and secant in quadrant IV; allied angles reduce any angle to an acute one with the correct sign.

Key allied-angle results:

- $\sin (-x)=-\sin x$, $\cos (-x)=\cos x$
- $\sin (\pi -x)=\sin x$, $\cos (\pi -x)=-\cos x$
- $\sin (\pi +x)=-\sin x$, $\cos (\pi +x)=-\cos x$
- $\sin (2\pi -x)=-\sin x$, $\cos (2\pi -x)=\cos x$
- $\sin \left(\frac{\pi }{2}+x\right)=\cos x$, $\cos \left(\frac{\pi }{2}+x\right)=-\sin x$

Worked examples.

$$\sin 150^{\circ }=\sin (180^{\circ }-30^{\circ })=\frac{1}{2}\cos 210^{\circ }=\cos (180^{\circ }+30^{\circ })=-\frac{\sqrt{3}}{2}$$
$$\tan 315^{\circ }=\tan (360^{\circ }-45^{\circ })=-1\cos 1110^{\circ }=\cos (3\times 360^{\circ }+30^{\circ })=\frac{\sqrt{3}}{2}$$
$$\sin \frac{31\pi }{3}=\sin \left(10\pi +\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}\cot \left(-\frac{15\pi }{4}\right)=\cot \left(-4\pi +\frac{\pi }{4}\right)=1$$

An everyday example. **A fan blade at $150^{\circ }$ is exactly as high as one at $30^{\circ }$, just on the other side — so their sines are equal.

The substance. With $\pi \pm x$ or $2\pi \pm x$ the function name stays; with $\frac{\pi }{2}\pm x$ it switches to the co-function.**

**Sine and cosine are defined for all real numbers, take values from $-1$ to $1$ and repeat every $2\pi$; tangent and cotangent repeat every $\pi$ and take all real values; secant and cosecant repeat every $2\pi$ and never lie strictly between $-1$ and $1$.

- $\sin x$, $\cos x$**: domain $\mathbb{R}$, range $[-1,1]$, period $2\pi$
- **$\tan x$**: domain $\mathbb{R}-\left\{(2n+1)\frac{\pi }{2}\right\}$, range $\mathbb{R}$, period $\pi$
- **$\cot x$**: domain $\mathbb{R}-\{n\pi \}$, range $\mathbb{R}$, period $\pi$
- **$\sec x$**: domain $\mathbb{R}-\left\{(2n+1)\frac{\pi }{2}\right\}$, range $(-\infty ,-1]\cup [1,\infty )$, period $2\pi$
- **$\mathrm{cosec}x$**: domain $\mathbb{R}-\{n\pi \}$, same range as $\sec x$, period $2\pi$

Graphs over one period.

- **$y=\sin x$** starts at $0$, rises to $1$ at $\frac{\pi }{2}$, returns to $0$ at $\pi$, falls to $-1$ at $\frac{3\pi }{2}$ and is back to $0$ at $2\pi$
- **$y=\cos x$** is the same wave shifted, starting at $1$
- **$y=\tan x$** rises through $0$ and shoots up towards vertical asymptotes at $\pm \frac{\pi }{2}$

An everyday example. The mains voltage supplied to Indian homes alternates like a sine wave, repeating $50$ times every second, so one period lasts $\frac{1}{50}$ s.

The substance. **Because the range of $\sin x$ is $[-1,1]$, an equation such as $\sin x=1.5$ has no solution.**