Step by step
The question
Question: find the exact value of sin 195°.
Exact means with roots and fractions, not a decimal.
1 Split the angle
195° is not on the unit circle.
But 135° and 60° are, and 135° plus 60° is 195°.
So sin 195° is sin of 135° plus 60°.
2 Write the rule
The sum rule for sine: sin(A + B) = sin A cos B + cos A sin B.
Careful!
We cannot split sin into two sins.
sin 135° + sin 60° is about 1.5731: more than 1.
A sine is never more than 1.
We use the rule.
A is 135° and B is 60°.
We put them in the rule.
3 Put in the values
From the unit circle: sin 135° = √2/2.
cos 60° = 1/2.
cos 135° = −√2/2.
sin 60° = √3/2.
135° is in quadrant II: the signs come from there.
We put the values in.
4 Multiply
Tops times tops, bottoms times bottoms.
A root times a root: √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.
5 Make one fraction
Both have the same bottom, 4.
So we put the tops together over 4: (√2 − √6)/4.
So sin 195° = (√2 − √6)/4.
6 Check
Let's check with the calculator.
Our answer: √2 is about 1.4142, √6 is about 2.4495.
Over 4: about −0.2588.
And sin 195° on the calculator is about −0.2588.
The same!
So it is right.
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.