Question

Find the exact value by using a sum or difference identity. 5) sin 165° A) -√√2(√√3-1) B)√√2(√√3-1) C) -√2(√3+1) D). √(√3-1) 5)

Answer 1

The exact value of sin 165° using a sum or difference identity is -√6(√6-√3)/4.

Step-by-step explanation:

To find the exact value of sin 165° using a sum or difference identity, we can use the identity sin(a + B) = sin a cos B + cos a sin B.

Let's choose a = 150° and B = 15°, since 150° + 15° = 165°.

Using the values in the identity, sin 165° = sin 150° cos 15° + cos 150° sin 15°

Simplifying the expression, sin 165° = (√3/2)(√6/4) + (-1/2)(√6/4)

Therefore, the exact value of sin 165° is (√3√6 - √6)/4, which is option A) -√6(√6-√3)/4.

Answer 2

The exact value of sin 165 is (√6 - √2)/4

In trigonometry there are special angles in which the their values can be exact. Some of this angles are;

30°, 45°, 60°

sin 30 = 1/2

cos 30 = √3/2

sin45 = cos 45 = 1/√2

sinθ = sin( 180 - θ)

therefore

sin 165 = sin ( 180 - 165)

sin 165 = sin 15

Sin 15 = sin (45 -30)

Sin ( A- B) = sin A cos B - cosAsinB

sin(45-30) = sin45cos30 - cos45sin30

= 1/√2 × √3/2) - (1/√2 × 1/2

= √3/2√2 - 1/2√2

= (√3-1)/2√2

= (2√6 - 2√2)/8

= (√6 - √2)/4