Trig Identities
\( \sin^2\theta + \cos^2\theta = 1 \)
Example
Simplify \(3(\sin x + \cos x)^2\)
\[
3(\sin x + \cos x)^2
\]
\[
= 3(\sin x + \cos x)(\sin x + \cos x)
\]
\[
= 3(\sin x \sin x + \sin x \cos x + \cos x \sin x + \cos x \cos x)
\]
\[
= 3(\sin^2 x + 2\sin x \cos x + \cos^2 x)
\]
\[
= 3\bigl(\,(\sin^2 x + \cos^2 x) + 2\sin x \cos x\,\bigr)
\]
\[
\text{Use } \sin^2 x + \cos^2 x = 1
\]
\[
= 3(1 + 2\sin x \cos x)
\]
\[
= 3 + 6\sin x \cos x
\]
This can be further processed:
\[
3\bigl(1 + 2\sin x \cos x\bigr)
\]
\[
\text{Use } \sin 2x = 2\sin x \cos x
\]
\[
= 3\bigl(1 + \sin 2x\bigr)
\]
\[
= 3 + 3\sin 2x
\]
\( \tan\theta = \dfrac{\sin\theta}{\cos\theta} \)
\[
\frac{\sin\alpha}{\cos\alpha}
= \frac{\frac{y}{r}}{\frac{x}{r}}
\]
\[
= \frac{y}{r} \times \frac{r}{x}
\]
\[
= \frac{y}{x}
\]
\[
\frac{y}{x}
= \frac{\text{Opposite}}{\text{Adjacent}}
= \tan\alpha
\]
Example
Express \(5\sin x \cos x \tan x\) in its simplest form.
Long Way
\[
5\sin x \cos x \tan x
\]
\[
\text{Use } \tan x = \frac{\sin x}{\cos x}
\]
\[
= 5\sin x \cos x \times \frac{\sin x}{\cos x}
\]
\[
= \frac{5\sin x \cos x}{1} \times \frac{\sin x}{\cos x}
\]
\[
= \frac{5\sin x \,\cancel{\cos x}}{1} \times \frac{\sin x}{\cancel{\cos x}}
\]
\[
= \frac{5\sin x \cdot \sin x}{1}
\]
\[
= 5\sin^2 x
\]
Quick way
\[
5\sin x\cos x\tan x
= 5\sin x\cos x\left(\frac{\sin x}{\cos x}\right)
= 5\sin^2 x
\]
Exact Values
\[
\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin 45^\circ = \frac{1}{\sqrt{2}}
\]
\[
\sin 45^\circ
= \frac{1}{\sqrt{2}}
\times \frac{\sqrt{2}}{\sqrt{2}}
= \frac{\sqrt{2}}{2}
\]
\[
\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
\cos 45^\circ = \frac{1}{\sqrt{2}}
\]
\[
\cos 45^\circ
= \frac{1}{\sqrt{2}}
\times \frac{\sqrt{2}}{\sqrt{2}}
= \frac{\sqrt{2}}{2}
\]
\[
\tan\theta = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan45^\circ = \frac{1}{1}
\]
\[
\tan 45^\circ
= 1
\]
\[
\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\]
\[
\sin 30^\circ
= \frac{1}{{2}}
\]
\[
\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
\cos 60^\circ = \frac{1}{{2}}
\]
\[
\cos 30^\circ
= \frac{\sqrt{3}}{2}
\]
\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan 60^\circ
= \frac{\sqrt{3}}{1}
= \sqrt{3}
\]
\[
\tan 30^\circ
= \frac{1}{\sqrt{3}}
= \frac{\sqrt{3}}{3}
\]
Learn :
Useful to know :
| Quadrant | Angle | Radians | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|---|---|
| Quadrant I | |||||
| I | 30° | \(\frac{\pi}{6}\) | \(\frac12\) | \(\frac{\sqrt3}{2}\) | \(\frac{\sqrt3}{3}\) |
| I | 45° | \(\frac{\pi}{4}\) | \(\frac{\sqrt2}{2}\) | \(\frac{\sqrt2}{2}\) | 1 |
| I | 60° | \(\frac{\pi}{3}\) | \(\frac{\sqrt3}{2}\) | \(\frac12\) | \(\sqrt3\) |
| Quadrant II | |||||
| II | 120° | \(\frac{2\pi}{3}\) | \(\frac{\sqrt3}{2}\) | \(-\frac12\) | \(-\sqrt3\) |
| II | 135° | \(\frac{3\pi}{4}\) | \(\frac{\sqrt2}{2}\) | \(-\frac{\sqrt2}{2}\) | \(-1\) |
| II | 150° | \(\frac{5\pi}{6}\) | \(\frac12\) | \(-\frac{\sqrt3}{2}\) | \(-\frac{\sqrt3}{3}\) |
| Quadrant III | |||||
| III | 210° | \(\frac{7\pi}{6}\) | \(-\frac12\) | \(-\frac{\sqrt3}{2}\) | \(\frac{\sqrt3}{3}\) |
| III | 225° | \(\frac{5\pi}{4}\) | \(-\frac{\sqrt2}{2}\) | \(-\frac{\sqrt2}{2}\) | 1 |
| III | 240° | \(\frac{4\pi}{3}\) | \(-\frac{\sqrt3}{2}\) | \(-\frac12\) | \(\sqrt3\) |
| Quadrant IV | |||||
| IV | 300° | \(\frac{5\pi}{3}\) | \(-\frac{\sqrt3}{2}\) | \(\frac12\) | \(-\sqrt3\) |
| IV | 315° | \(\frac{7\pi}{4}\) | \(-\frac{\sqrt2}{2}\) | \(\frac{\sqrt2}{2}\) | \(-1\) |
| IV | 330° | \(\frac{11\pi}{6}\) | \(-\frac12\) | \(\frac{\sqrt3}{2}\) | \(-\frac{\sqrt3}{3}\) |
Other Trig Facts
Complementary angles
\[
\sin(\alpha)
= \frac{y}{r}
\]
\[
\cos(\alpha)
= \frac{x}{r}
\]
\[
\sin(90^\circ-\alpha)
= \frac{x}{r}
= \cos\alpha
\]
\[
\cos(90^\circ-\alpha)
= \frac{y}{r}
= \sin\alpha
\]
\[
\sin(90^\circ-\alpha) = \cos\alpha
\]
\[
\cos(90^\circ-\alpha) = \sin\alpha
\]
Supplementary angles
\[
\sin(180^\circ-\alpha)
= \frac{y}{r}
= \sin\alpha
\]
\[
\cos(180^\circ-\alpha)
= \frac{-x}{r}
= -\cos\alpha
\]
\[
\sin(180^\circ-\alpha) = \sin\alpha
\]
\[
\cos(180^\circ-\alpha) = - \cos\alpha
\]
Negative angles
\[
\sin(-\alpha)
= \frac{-y}{r}
= -\sin\alpha
\]
\[
\cos(-\alpha)
= \frac{x}{r}
= \cos\alpha
\]
\[
\sin(-\alpha) = -\sin\alpha
\]
\[
\cos(-\alpha) = \cos\alpha
\]
