Trigonometry Proofs

$$\cos^2a+\sin^2a=1$$

pick point on unit circle (CosA, SinA)

use distance formula between point and origin

Sine Rule

See triangle on page 16

Use area formula from each angle and let equal

Cosine Rule

draw a circle with radius b

choose point on circle and get coordinates using SOHCAHTOA

label a point on x-axis (c, 0)

label distance between points ‘a’

use distance formula and multiply out

$$\cos\left(a-b\right)=\cos a\cos b+\sin a\sin b$$

draw unit circle

label point Y (cosA, sinA)

label point X (cosB, sinB)

draw triangle using info

use cosine rule for |XY|²

use distance formula for |XY|²

let equal

$$\cos\left(a+b\right)=\cos a\cos b-\sin a\sin b$$

use same proof as cos(a-b)

sub -b in for b and use sin(-a)=sina

$$\cos2a=\cos^2a-\sin^2a$$

use cos(a+b) and sub in a for b

$$\sin\left(a+b\right)=\sin a\cos b+\cos a\sin b$$

use cos(a-b) identity

sub 90-A in for a

remember sinA=cos(90-A) vice versa

$$\tan\left(a+b\right)$$

sub in a+b for a

use compound angles to rewrite sin and cos

divide each term by cosAcosB

sub in tanA