Rewrite 23π12 as 2π- π12. Then, use the Angle Difference Formula for a sine. To find the value of the sine of π12, rewrite π12 as π3- π4. Then, use the Angle Difference Formula.
sqrt(2)-sqrt(6)/4
Practice makes perfect
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 sine.
sin ( A- B)=sin Acos B-cos 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.
sin 2π cos π/12- cos 2π sin π/12
=
( 0) cos π/12-( 1) sin π/12
Let's simplify the obtained expression!
Therefore, we found that sin 23π12 = - sin π12. Be aware that π12 is the difference of π3 and π4. Therefore, we can rewrite sin π12 as the sine of a difference.
sin π/12=sin (π/3-π/4)
We can once again use the Sine 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.
sin π/3 cos π/4- cos π/3 sin π/4
=
( sqrt(3)/2) sqrt(2)/2-( 1/2) sqrt(2)/2
Let's finally simplify the obtained expression!
