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Unit 11 Review: Trig Identities
NAME: ______________________________________
Directions for 1: Simplify to a single Directions for 2-‐4: Prove the identity. SHOW WORK! trig ratio or number. 1) !"#! !"!!
!"#! 2) 1 + 𝑠𝑒𝑐! 𝑥 𝑠𝑖𝑛! 𝑥 = 𝑠𝑒𝑐! 𝑥
3) !"#!!!!"#!!!
+ !"#!!!!"#!!!
= 0 4) sin 2𝛼 (cot𝛼 + tan𝛼) = 2 Directions for 5-‐6: Use the sum/difference/double or half angle formulas to find the exact value. 5) cos 22.5° 6) sin 255°
Sum/Difference Identities
sin(𝛼 ± 𝛽) = sin𝛼 cos 𝛽 ± sin𝛽 cos 𝛼
cos(𝛼 ± 𝛽) = cos𝛼 cos 𝛽 ∓ sin𝛼 sin𝛽
tan(𝛼 ± 𝛽) =tan𝛼 ± tan𝛽1 ∓ tan𝛼 tan𝛽
Double Angle Formulas
sin 2𝜃 = 2 sin𝜃 cos𝜃
cos 2𝜃 = 𝑐𝑜𝑠! 𝜃 − 𝑠𝑖𝑛! 𝜃
= 2 𝑐𝑜𝑠! 𝜃 − 1
= 1 − 2𝑠𝑖𝑛! 𝜃
tan 2𝜃 =2 tan 𝜃
1 − 𝑡𝑎𝑛! 𝜃
Half Angle Identities
sin𝑥2= ±!
1 − cos 𝑥2
cos𝑥2= ±!
1 + cos 𝑥2
tan𝑥2= ±!
1 − cos 𝑥1 + cos 𝑥
Pythagorean Identities
𝑠𝑖𝑛! 𝜃 + 𝑐𝑜𝑠! 𝜃 = 1
𝑡𝑎𝑛! 𝜃 + 1 = 𝑠𝑒𝑐! 𝜃
1 + 𝑐𝑜𝑡! 𝜃 = 𝑐𝑠𝑐! 𝜃
Directions for 7-‐8: If tan 𝑥 = − !"! and 𝑥 is in Quadrant II and sin 𝑦 = − !
! and y is in Quadrant IV, find the exact value. Draw
the reference triangle.
7) cos (𝑥 − 𝑦) 8) sin !!
Directions for 9: Find the exact value of Directions for 10: Find the approximate value the following where °≤≤° 3600 x . of the following where °≤≤° 3600 x . 9) 𝑡𝑎𝑛! 𝑥 − 3 tan 𝑥 = 0 10) 10 sin 𝑥 + 15 = 20 − 3 sin 𝑥
Applications/Extensions
1) For each equation use your calculator to determine if the following are identities. Sketch the graph (with zoom trig) to show they are or aren’t identities. If the equation is an identity, VERIFY IT! If it is not an identity, find a value of
x, 𝑥 = 0,𝜋, !!, !!, !! for which both sides are defined but not equal.
a) !!!"#!!"#!
= tan 𝑥 b) !!!"#!!!!"#!
= !"#!!"#!
Sketch: Sketch: Verification or value of x it does not Verification or value of x it does not work for (SHOW WORK: work for (SHOW WORK: