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…

Sum and Difference Identities Make note of the signs!!

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°)= 2 2 ∙ 3 2 − 2 2 ∙ 1 2 = 6 4 − 2 4 = 6 − 2 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°)= 2 2 ∙ −1 2 − 2 2 ∙ 3 2 = − 2 4 − 6 4 = − 2 − 6 4

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°)= 1+ 3 3 1− 3 3 Multiply by 3 3 = 3+ 3 3− 3