Calculate sin(15°) using any half-angle identity. You must express your answer as an exact value.

Question

This problem can be solved using the sin() half angle trig identity.

Answer 1

There is another useful half angle formula

for y=sin(x/2)

which is derived by the two equation system

(sin(x/2)+cos(x/2))²=1+sin(x)

(sin(x/2)-cos(x-/2))²=1-sin(x)

so sin(x/2)=(1/2)*[+/-√(1+sin(x))+/-√(1-sin(x))]

since sin(30) equals 1/2

sin(15)=(1/2)*[+/-√(3/2)+/-√(1/2)]

need sin(15)>0 and sin(15)<1/2

so choice must be + for first term and - for second term

sin(15)=(1/2)*[√(3/2)-√(1/2)]

=[(√6-√2)/4

it is worth noting, the same two equation system

resolved by subtracting versus adding equations

results in half angle formula for cos(x/2)

cos(x/2)=(1/2)*[+/-√(1+sin(x))+/-√(1-(sin(x))]

since cos(15)>cos(30)>0

would choose + and +

cos(15)=(1/2)*[√(3/2)+√(1/2)]

Answer 2

 The identity is as follows:

$$sin(\frac{\Theta}{2})=\pm\sqrt{\frac{1-cos(\Theta)}{2}}$$

If we substitute 30° for $$\Theta$$, then we get

$$sin(\frac{30}{2})=\pm\sqrt{\frac{1-cos(30)}{2}}$$

This can be simplified to

$$sin(15)=\pm\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}$$

Splitting up the numerator of the fraction results in:

$$sin(15)=\pm\sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$$

We can make the make the first fraction 2/4, and the two fractions which have 4 as a common denominator:

$$sin(15)=\pm\sqrt{\frac{2-\sqrt{3}}{4}}$$

This is an acceptable answer, and using advanced algebraic manipulation techniques, this answer can written as:

sin(15)= $$\frac{\sqrt{6}-\sqrt{2}}{4}$$