How do you use the sum and difference formula to simplify #sin165#?

Question

Brooklyn Anderson · Accepted Answer

Use sin(135+30).

The sum formula for sin is #sin(A+B)=sinAcosB+cosAsinB# Choose #A=135# and #B=30# because both of these values lie on the unit circle. From the unit circle: #cos135=-sqrt2/2# #sin135=sqrt2/2# #cos30=sqrt3/2# #sin30=1/2# Let #A=135# and #B=30# Substitute. #sin(165)=sin(135+30)=sqrt2/2*sqrt3/2 + -sqrt2/2 * 1/2# #sin(165)= sqrt6/4 -sqrt2/4# #sin(165)=(sqrt6-sqrt2)/4#

Charles Autterson · Answer

undefined To simplify $\sin(165^\circ)$ using the sum and difference formula for sine, we express $165^\circ$ as the sum or difference of known angles whose sine values we can calculate easily.

Since $165^\circ = 180^\circ - 15^\circ$, we can use the difference formula for sine:

$$
\sin(A - B) = \sin A \cos B - \cos A \sin B
$$

Let $A = 180^\circ$ and $B = 15^\circ$. We know that $\sin(180^\circ) = 0$ and $\cos(180^\circ) = -1$, and $\sin(15^\circ)$ and $\cos(15^\circ)$ can be calculated.

Substituting these values into the formula:

$$
\sin(165^\circ) = \sin(180^\circ - 15^\circ) = \sin 180^\circ \cos 15^\circ - \cos 180^\circ \sin 15^\circ
$$

$$
= (0)(\cos 15^\circ) - (-1)(\sin 15^\circ) = \sin 15^\circ
$$

Now, you can use the value of $\sin 15^\circ$ obtained from trigonometric tables or calculated using trigonometric identities.