Trigonometric Identities and Equations

Starting point: Years 9 and 10

Solving a simple trig equation

To solve $\sin \theta =0.6$ for an acute angle, use the inverse:

$$\theta ={\sin }^{-1}(0.6)\approx 36.9^{\circ }$$

Rearrange first if needed: $4\cos \theta =3$ gives $\cos \theta =0.75$, so $\theta \approx 41.4^{\circ }$.

Two identities from one triangle

Take a right triangle with hypotenuse 1 and angle $\theta$. The opposite side is $\sin \theta$ and the adjacent side is $\cos \theta$.

Opposite over adjacent gives

$$\tan \theta =\frac{\sin \theta }{\cos \theta }$$

Pythagoras gives

$${\sin }^2\theta +{\cos }^2\theta =1$$

Here ${\sin }^2\theta$ means $(\sin \theta {)}^2$.

Quick summary

• $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and ${\sin }^2\theta +{\cos }^2\theta =1$.
• Reciprocal ratios: $\csc$, $\sec$, $\cot$. Also $1+{\tan }^2\theta ={\sec }^2\theta$.
• Prove identities by working on one side, usually changing to sin and cos.
• To solve: isolate the ratio, find the related angle, use ASTC for all quadrants, check the domain.
• For $\sin bx$, widen the domain by a factor of $b$ before solving.

Quiz