Worked examples
The question
Question: find the exact value of sin 75°.
Exact means with roots and fractions, not a decimal.
sin 75° =
1 Split the angle
75° is not on the unit circle.
But 45° and 30° are, and 45° plus 30° is 75°.
75° = 45° + 30°
So sin 75° is sin of 45° plus 30°.
= sin(45° + 30°)
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.
sin 45° + sin 30° is about 1.2071: more than 1.
A sine is never more than 1.
We use the rule.
Careful: not sin 75° = sin 45° + sin 30° ✗
A is 45° and B is 30°.
We put them in the rule.
= sin 45° cos 30° + cos 45° sin 30°
3 Put in the values
From the unit circle: sin 45° = √2/2.
cos 30° = √3/2.
cos 45° = √2/2.
sin 30° = 1/2.
sin 45° = √22, cos 45° = √22
cos 30° = √32, sin 30° = 12
We put the values in.
= √22 × √32 + √22 × 12
4 Multiply
Tops times tops, bottoms times bottoms.
A root times a root: √2 × √3 = √6.
√2 × √3 = √6
√2/2 times √3/2 is √6/4.
√2/2 times 1/2 is √2/4.
= √64 + √24
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 75° = (√6 + √2)/4.
sin 75° = √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 75° on the calculator is about 0.9659.
The same!
So it is right.
sin 75° ≈ 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 sin 75° = √6 + √24
▶ Watch it solved
The question
Question: find the exact value of sin 165°.
Exact means with roots and fractions, not a decimal.
sin 165° =
1 Split the angle
165° is not on the unit circle.
But 120° and 45° are, and 120° plus 45° is 165°.
165° = 120° + 45°
So sin 165° is sin of 120° plus 45°.
= sin(120° + 45°)
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.
sin 120° + sin 45° is about 1.5731: more than 1.
A sine is never more than 1.
We use the rule.
Careful: not sin 165° = sin 120° + sin 45° ✗
A is 120° and B is 45°.
We put them in the rule.
= sin 120° cos 45° + cos 120° sin 45°
3 Put in the values
From the unit circle: sin 120° = √3/2.
cos 45° = √2/2.
cos 120° = −1/2.
sin 45° = √2/2.
120° is in quadrant II: the signs come from there.
sin 120° = √32, cos 120° = −12
cos 45° = √22, sin 45° = √22
We put the values in.
= √32 × √22 + (−12 × √22)
4 Multiply
Tops times tops, bottoms times bottoms.
A root times a root: √3 × √2 = √6.
√3 × √2 = √6
√3/2 times √2/2 is √6/4.
−1/2 times √2/2 is −√2/4.
Adding a negative is taking away.
= √64 − √24
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 165° = (√6 − √2)/4.
sin 165° = √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 0.2588.
√6 − √24 ≈ (2.4495 − 1.4142) ÷ 4 ≈ 0.2588
And sin 165° on the calculator is about 0.2588.
The same!
So it is right.
sin 165° ≈ 0.2588
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 sin 165° = √6 − √24
▶ Watch it solved
The question
Question: find the exact value of sin 50° cos 10° + cos 50° sin 10°.
50° and 10° are not on the unit circle.
sin 50° cos 10° + cos 50° sin 10°
1 Spot the rule
This is the right side of a rule we know: sin(A + B) = sin A cos B + cos A sin B.
With A = 50° and B = 10°.
sin(A + B) = sin A cos B + cos A sin B
Careful!
The angle in the first sin is A.
So A is 50° and B is 10°.
The order matters when we take away.
Careful: sin first: sin A cos B, so A = 50°
So it is all one sin: sin(50° + 10°).
= sin(50° + 10°)
2 Add the angles
50° plus 10° is 60°.
That angle is on the unit circle!
= sin 60°
3 Use the unit circle
From the unit circle: sin 60° = √3/2.
= √32
4 Check
Let's check with the calculator.
Each one rounded: the whole thing is about 0.8660.
0.7660 × 0.9848 + 0.6428 × 0.1736 ≈ 0.8660
And √3/2 is about 0.8660.
The same!
So it is right.
√32 ≈ 0.8660
Let's look back.
First: spot the rule.
Then: add the angles.
Then: use the unit circle.
Last: check.
▶ Watch it solved
