Question

Use a sum or difference identity to find an exact value sin(15 degrees)

Answer 1

15 is half of 30 so use half angle formula for sine

Answer 2

This will be a rather long story...

Note: everything becomes a lot clearer if you write this out yourself,
with proper rationals, and not everything on one and the same line...

30 + 15 = 45.
We know the exact values of sin and cos for 30 and 45 degrees.
Also the formulas for sum of angles are

sin 45 = sin(30+15) = sin30cos15 + cos30sin15. (i)
cos 45 = cos(30+15) = cos30cos15 - sin30sin15. (ii)

Now, sin45 = cos45 (= sqrt(2)/2), so (i) = (ii).

Also, sin30 = 1/2 and cos30 = sqrt(3)/2.

Putting these values in (i) and (ii) gives:

1/2 cos15 + sqrt(3)/2 sin15 = sqrt(3)/2 cos15 - 1/2 sin15.

Collect terms with sin15 and with cos15:

(sqrt(3)/2 + 1/2)sin15 = (sqrt(3)/2 - 1/2)cos15.

Divide by sqrt(3)/2 + 1/2:

sin15 = (sqrt(3)/2 - 1/2)/(sqrt(3)/2 + 1/2) * cos15.

Isolate the factor 1/2:

sin15 = (1/2 *(sqrt(3) - 1))/(1/2 * (sqrt(3) + 1) * cos15.

Now get rid of the 1/2:

sin15 = (sqrt(3) - 1)/(sqrt(3) +1) * cos15. (iii)

We seem to be stuck at this point, because now we also need cos15 :(.
Now remember the formula sin2x=2sinxcosx.
This means that sin30=2sin15*cos15.

So cos15 = sin30/(2sin15) = 1/2/(2sin15) = 1/(4sin15).

We can use this result in (iii) and get:

sin15 = (sqrt(3) - 1)/(sqrt(3) + 1) * (1/(4sin15)).

Multiply both sides with 4sin15:

4(sin15)^2 = (sqrt(3) - 1)/(sqrt(3) + 1). (iv)

Now it is time to "simplify" the right side of (iv),
by multiplying both nominator and denominator with sqrt(3) - 1.

4(sin15)^2 = (sqrt(3) - 1)^2 / ((sqrt(3) + 1)(sqrt(3) - 1))
4(sin15)^2 = (3 - 2sqrt(3) +1) / (3 - 1) = (4 - 2sqrt(3)) / 2 = 2 - sqrt(3). (v)

Hold on, we are almost there!

In (v), divide by 4 and take the root (sin15 is in the first quadrant, so it is positive and
we do not need to worry about a negative value...).
This gives:

sin15 = sqrt((2 - sqrt(3))/4).

Taking the root of 4 gives, at last,

sin15 = sqrt(2 - sqrt(3)) / 2


Hope this helps!

ZeHanz

Answer 3

\[\sin(15°)\]\[=\sin(45°-30°)\]
\[\qquad\qquad\qquad \boxed{\sin(u\pm v)=\sin(u)\cos(v)\pm \cos(u)\sin(v)}\]
\[=\sin(45°)\cos(30°)-\cos(45°)\sin(30°)\]





\[=\frac{\sqrt 2}2\cdot\frac{\sqrt3}2-\frac{\sqrt 2}2\cdot\frac12\]\[=\frac{\sqrt2(\sqrt 3-1)}4\]