Question

find the exact value of the expression given below
cos (105°)
rewrite the expression using a sum or difference formula. choose the correct answer below
oa. cos (105°)=cos (60° + 45°)= sin (60°)cos (45°)- cos (60°)sin (45°)
ob. cos (105°)=cos (60° + 45°)= sin (60°)cos (45°)+ cos (60°)sin (45°)
oc. cos (105°)=cos (60° + 45°)= cos (60°)cos (45°)+ sin (60°)sin (45°)
od. cos (105°)=cos (60° + 45°)= cos (60°)cos (45°)- sin (60°)sin (45°)

Explanation:

Step1: Recall the cosine sum formula

The formula for \(\cos(A + B)\) is \(\cos(A)\cos(B)-\sin(A)\sin(B)\).

Step2: Identify \(A\) and \(B\)

Here \(A = 60^{\circ}\) and \(B=45^{\circ}\), so \(\cos(105^{\circ})=\cos(60^{\circ}+ 45^{\circ})\).
Using the formula \(\cos(A + B)=\cos(A)\cos(B)-\sin(A)\sin(B)\), we substitute \(A = 60^{\circ}\) and \(B = 45^{\circ}\) to get \(\cos(60^{\circ}+45^{\circ})=\cos(60^{\circ})\cos(45^{\circ})-\sin(60^{\circ})\sin(45^{\circ})\).

Answer:

D. \(\cos(105^{\circ})=\cos(60^{\circ}+45^{\circ})=\cos(60^{\circ})\cos(45^{\circ})-\sin(60^{\circ})\sin(45^{\circ})\)