use the angle sum or difference identity to find the exact value of sin 15 degrees

Question

I don't understand what to do after i the two numbers like (45-30)

Answer 1

Find the EXACT value of sin(15°) using a sum or difference formula.

 

In this solution the following will be strictly adhered to:

1. Trig Functions are FUNCTIONS and proper functional notation should always be used.

2. Parentheses will be used to create non-ambiguous expressions.

3. The value will be expressed as requested; i.e., in EXACT form (NO CALCULATORS WILL BE USED).

 

sin(15°) = sin(45° – 30°)

= sin(45°)cos(30°) – cos(45°)sin(30°)

 

Recall the two Special Triangles: 45°-45°-90° and 30°-60°-90°.

Their sides across from the angles are in the ratios: 1:1:√(2) & 1:√(3):2.

Draw and label diagrams of the two Special Triangles to guide you.

 

sin(15°) = sin(45°)cos(30°) – cos(45°)sin(30°)

 

= (1/√(2))(√(3)/2) – (1/√(2))(1/2)

 

= (1/√(2)) ( (√(3)/2) – (1/2) )

 

= (√(2)/2) ( (√(3) – 1)/2) )

 

= ( √(6) – √(2) )/4

Answer 2

sin(45-30) = sin45cos30 - cos45sin30 = (1/√2    x  √3/2)  - (1/2   x  1/2) = √3/2√2 - 1/4

:)

Answer 3

sin(15)=sin(45-30)

 

knowing the following equation:

 

sin(α – β) = sin(α)cos(β) – cos(α)sin(β)

 

we can say:

 

sin(45-30)=sin45*cos30 - cos45*sin30

 

sin45=cos45=2^0.5/2

sin30=1/2, cos30=3^0.5/2

 

plugging in these values:

 

sin(45-30)=(2^0.5/2*3^0.5/2)-(2^0.5/2*1/2)

 

doing the calculations:

 

sin(15) = 0.259