Question

Use a sum or difference identity to find the exact value

sin 10 cos 110 + cos 10 sin 110

A. 1/6
B. √3/2
C. -√3/2
D. -√1/2

Answer 1

Using the sum identity for sine, sin(10 + 110) simplifies to sin(120). Since 120 degrees corresponds to a standard position angle in a 30-60-90 triangle, the exact value is √3/2.

Step-by-step explanation:

To find the exact value of sin 10 cos 110 + cos 10 sin 110, we will use a sum identity for sine. Specifically, we use the identity sin(a ± β) = sin a cos β ± cos a sin β.

Applying this identity:

• sin(10 + 110) = sin(10)cos(110) + cos(10)sin(110)

Now, sin(10 + 110) simplifies to sin(120), which is an angle we can evaluate exactly because 120 degrees corresponds to a standard position angle in a 30-60-90 triangle.

The sine of 120 degrees, which is in the second quadrant where sine is positive, is the same as the sine of 60 degrees due to symmetry, giving us √3/2.