1. Review
Find the EXACT value of: 1. sin 30° 2. cos 225° 3. tan 135° 4. cos 300° How can we find the values of expressions like sin 15° ?? We need some new formulas…

2. Sum and Difference Identities
Make note of the signs!!

3. Evaluate: sin 15° sin 15°= sin(45°– 30°)
Rewrite the problem as the sum or difference of two angles you DO know the sine value of (i.e. off the unit circle):
NOTE: You could use sin(6𝟎°– 4𝟓°) here and would get the same answer.
sin 15°= sin(45°– 30°)
Use the sum/difference formula to evaluate:
sin(45°– 30°)= sin(45°)cos(30°) – cos(45°)sin(30°)
Simplify (NO CALCULATOR):
You can check your answer by converting your answer to a decimal and comparing it to the decimal value you get when you enter “sin15” in your calculator.
sin(45°– 30°)= ∙ − ∙ 1 2
= −
= 6 − 2 4

4. Evaluate: cos 165° cos 165°= cos(45°+ 120°)
Rewrite the problem as the sum or difference of two angles you DO know the cosine value of (i.e. off the unit circle):
NOTE: You could use cos(210°– 4𝟓°) here and would get the same answer.
cos 165°= cos(45°+ 120°)
Use the sum/difference formula to evaluate:
cos(45°+120°)= cos(45°)cos(120°) – sin(45°)sin(120°)
Simplify (NO CALCULATOR):
You can check your answer by converting your answer to a decimal and comparing it to the decimal value you get when you enter “cos165” in your calculator.
cos(45°+120°)= ∙ −1 2 − ∙
= − −
= − 2 − 6 4

5. Evaluate: tan 75° tan 75°= tan(45°+ 30°)
Rewrite the problem as the sum or difference of two angles you DO know the tangent value of (i.e. off the unit circle):
NOTE: You could use tan(120°– 4𝟓°) here and would get the same answer.
tan 75°= tan(45°+ 30°)
Use the sum/difference formula to evaluate:
tan(45°+30°)= tan 45°+ tan 30° 1− tan 45°∙ tan 30°
Simplify (NO CALCULATOR):
You can check your answer by converting your answer to a decimal and comparing it to the decimal value you get when you enter “cos165” in your calculator.
tan(45°+30°)= −
Multiply by 3 3
= − 3