Before we rewrite the given expression, let's start by recalling the values of the three main trigonometric functions for the most important angles.
sin θ
cos θ
tan θ
θ =0
0
1
0
θ =π/6
1/2
sqrt(3)/2
sqrt(3)/3
θ =π/4
sqrt(2)/2
sqrt(2)/2
1
θ =π/3
sqrt(3)/2
1/2
sqrt(3)
θ =π/2
1
0
-
θ =π
0
- 1
0
θ =2π
0
1
0
Let's now recall the Angle Difference Formula for a cosine function.
cos(A-B)=cos A cos B+sin A sin B
We will use this identity to rewrite the given expression.
Next, we will use the table we constructed at the beginning of this solution to simplify this expression.
cos π cos π/12 - sin π sin π/12
=
( - 1) cos π/12-( 0) sin π/12
Let's simplify the obtained expression!
Therefore, we found that cos 11π12 = - cos π12. Be aware that π12 is the difference of π3 and π4. Therefore, we can rewrite cos π12 as the cosine of a difference.
cos π/12=cos (π/3-π/4)
We can once again use the Cosine Difference Formula to find the exact value of the expression.
Next, we will use the table we constructed at the beginning of this solution to simplify this expression.
cos π/3 cos π/4+ sin π/3 sin π/4
=
( 1/2) sqrt(2)/2+( sqrt(3)/2) sqrt(2)/2
Let's finally simplify the obtained expression!
