Trigonometry/Trigonometric Formula Reference

Exercise: Remembering Formulas Cover the right hand side of each formula, and use the information about remembering formulas from the previous page to get the right hand side.

Principal Trig Relationships

The following identities give relationships between the trigonometric functions.

1. $\sin x =\cos\left(\frac{\pi}{2}-x\right)\quad \,$
2. $\cos x =\sin\left(\frac{\pi}{2}-x\right)\quad \,$
3. $\displaystyle\tan x =\frac{\sin x}{\cos x}$
4. $\displaystyle\csc x = \frac{1}{\sin x}$
5. $\displaystyle\sec x = \frac{1}{\cos x}$

Pythagoras related

1. $\displaystyle\sin^2{\theta}+\cos^2{\theta}=1$
2. $\displaystyle\tan^2{\theta}+1=\sec^2{\theta}$

Worked Example: The cot formula is missing One formula is missing. By dividing the $\displaystyle\sin^2{\theta}+\cos^2{\theta}=1$ by $\displaystyle\sin^2{\theta}$ or by $\displaystyle\cos^2{\theta}$ we can get two other formulas. The missing formula is obtained by dividing through by $\displaystyle\sin^2{\theta}$ $\displaystyle\frac{\sin^2{\theta}}{\sin^2{\theta}}+\frac{\cos^2{\theta}}{\sin^2{\theta}}=\frac{1}{\sin^2{\theta}}$ The missing formula is: $\displaystyle 1+\cot^2{\theta}=\csc^2{\theta}$

Periodicity

Four trigonometric functions are $\displaystyle2\pi$ periodic:

1. $\displaystyle\sin{\theta}=\sin{(\theta+2\pi)}$
2. $\displaystyle\cos{\theta}=\cos{(\theta+2\pi)}$
3. $\displaystyle\csc{\theta}=\csc{(\theta+2\pi)}$
4. $\displaystyle\sec{\theta}=\sec{(\theta+2\pi)}$

Two trigonometric functions are $\displaystyle\pi$ periodic:

1. $\displaystyle\tan{\theta}=\tan{(\theta+\pi)}$
2. $\displaystyle\cot{\theta}=\cot{(\theta+\pi)}$

Angle Sums

Formulas involving sums of angles are as follows:

1. $\displaystyle\sin{(\alpha+\beta)}=\sin{\alpha}\cos{\beta}+\cos{\alpha}\sin{\beta}$
2. $\displaystyle\cos{(\alpha+\beta)}=\cos{\alpha}\cos{\beta}-\sin{\alpha}\sin{\beta}$
3. $\displaystyle\sin{(\alpha-\beta)}=\sin{\alpha}\cos{\beta}-\cos{\alpha}\sin{\beta}$
4. $\displaystyle\cos{(\alpha-\beta)}=\cos{\alpha}\cos{\beta}+\sin{\alpha}\sin{\beta}$

Multiple Angle Formulas

Substituting $\displaystyle\beta=\alpha$ gives the double angle formulae

1. $\displaystyle\sin{(2\alpha)}=2\sin({\alpha})\cos({\alpha})$
2. $\displaystyle\cos{(2\alpha)}=\cos^2{\alpha}-\sin^2{\alpha}$

Substituting $\displaystyle\sin^2{\alpha}+\cos^2{\alpha}=1$ gives

1. $\displaystyle\cos{(2\alpha)}=2\cos^2{\alpha}-1$
2. $\displaystyle\cos{(2\alpha)}=1-2\sin^2{\alpha}$

These can be obtained by putting $\displaystyle\beta=2\theta, \alpha=\theta$ in the addition formula.

1. $\displaystyle\sin{(3\theta)}=3\sin{\theta}-4\sin^3{\theta}$
2. $\displaystyle\cos{(3\theta)}=4\cos^3{\theta}-3\cos{\theta}$
3. $\displaystyle\tan{(3\theta)}=\frac{3\tan{\theta}-\tan^3{\theta}}{1-3\tan^2{\theta}}$
   
   

This can also be obtained from the angle sums formula.

1. $\displaystyle2\sin{(A)}\cos{(B)}=\sin(A+B)+\sin(A-B)$

Trigonometric functions of some closely related angles

This list may duplicate some of the periodicity formulas above, but all the formulas are given for the sake of completeness. Angles are expressed in degrees rather than radians. Similar relations for cot, sec and cosec follow immediately from the definitions of these functions; just replace sin by cosec, cos by sec and tan by cot (and vice versa).

sin(x)

1. $\displaystyle\sin(-x) = -\sin(x)$
2. $\displaystyle\sin(90^\circ-x) = \cos(x)$
3. $\displaystyle\sin(90^\circ+x) = \cos(x)$
4. $\displaystyle\sin(180^\circ-x) = \sin(x)$
5. $\displaystyle\sin(180^\circ+x) = -\sin(x)$
6. $\displaystyle\sin(270^\circ-x) = -\cos(x)$
7. $\displaystyle\sin(270^\circ+x) = -\cos(x)$
8. $\displaystyle\sin(360^\circ-x) = -\sin(x)$
9. $\displaystyle\sin(360^\circ+x) = \sin(x)$

cos(x)

1. $\displaystyle\cos(-x) = \cos(x)$
2. $\displaystyle\cos(90^\circ-x) = \sin(x)$
3. $\displaystyle\cos(90^\circ+x) = -\sin(x)$
4. $\displaystyle\cos(180^\circ-x) = -\cos(x)$
5. $\displaystyle\cos(180^\circ+x) = -\cos(x)$
6. $\displaystyle\cos(270^\circ-x) = -\sin(x)$
7. $\displaystyle\cos(270^\circ+x) = \sin(x)$
8. $\displaystyle\cos(360^\circ-x) = \cos(x)$
9. $\displaystyle\cos(360^\circ+x) = \cos(x)$

tan(x)

1. $\displaystyle\tan(-x) = -\tan(x)$
2. $\displaystyle\tan(90^\circ-x) = \cot(x)$
3. $\displaystyle\tan(90^\circ+x) = -\cot(x)$
4. $\displaystyle\tan(180^\circ-x) = -\tan(x)$
5. $\displaystyle\tan(180^\circ+x) = \tan(x)$
6. $\displaystyle\tan(270^\circ-x) = \cot(x)$
7. $\displaystyle\tan(270^\circ+x) = -\cot(x)$
8. $\displaystyle\tan(360^\circ-x) = -\tan(x)$
9. $\displaystyle\tan(360^\circ+x) = \tan(x)$

Exercise: Radians Rewrite the above formulas using radians.