Exact value of sin 285° Step by Step | Sum and difference identities

Step by step

The question

Question: find the exact value of sin 285°.

Exact means with roots and fractions, not a decimal.

sin 285° =

1 Split the angle

285° is not on the unit circle.

But 225° and 60° are, and 225° plus 60° is 285°.

285° = 225° + 60°

So sin 285° is sin of 225° plus 60°.

= sin(225° + 60°)

2 Write the rule

The sum rule for sine: sin(A + B) = sin A cos B + cos A sin B.

sin(A + B) = sin A cos B + cos A sin B

Careful!

We cannot split sin into two sins.

We use the rule.

Careful: not sin 285° = sin 225° + sin 60° ✗

A is 225° and B is 60°.

We put them in the rule.

= sin 225° cos 60° + cos 225° sin 60°

3 Put in the values

From the unit circle: sin 225° = −√2/2.

cos 60° = 1/2.

cos 225° = −√2/2.

sin 60° = √3/2.

225° is in quadrant III: the signs come from there.

sin 225° = −√22, cos 225° = −√22

cos 60° = 12, sin 60° = √32

We put the values in.

= −√22 × 12 + (−√22 × √32)

4 Multiply

Tops times tops, bottoms times bottoms.

A root times a root: √2 × √3 = √6.

√2 × √3 = √6

−√2/2 times 1/2 is −√2/4.

−√2/2 times √3/2 is −√6/4.

Adding a negative is taking away.

= −√24 − √64

5 Make one fraction

Both have the same bottom, 4.

So we put the tops together over 4: (−√6 − √2)/4.

= −√6 − √24

So sin 285° = (−√6 − √2)/4.

sin 285° = −√6 − √24

6 Check

Let's check with the calculator.

Our answer: √6 is about 2.4495, √2 is about 1.4142.

Over 4: about −0.9659.

−√6 − √24 ≈ (−2.4495 − 1.4142) ÷ 4 ≈ −0.9659

And sin 285° on the calculator is about −0.9659.

The same!

So it is right.

sin 285° ≈ −0.9659

Let's look back.

First: split the angle.

Then: write the rule.

Then: put in the values.

Then: multiply.

Then: make one fraction.

Last: check.

Answer