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Trigonometric Identities Formulas
Pythagorean Identities sin2 𝜃 + cos2 𝜃 = 1
1 + cot2 𝜃 = csc2 𝜃
tan2 𝜃 + 1 = sec2 𝜃
Reciprocal Identities
𝑠𝑖𝑛𝜃 =1
𝑐𝑠𝑐𝜃
𝑐𝑠𝑐𝜃 =
1𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃 =1
𝑠𝑒𝑐𝜃
𝑠𝑒𝑐𝜃 =
1𝑐𝑜𝑠𝜃
𝑡𝑎𝑛𝜃 =1
𝑐𝑜𝑡𝜃
𝑐𝑜𝑡𝜃 =
1𝑡𝑎𝑛𝜃
Quotient Identities
𝑡𝑎𝑛𝜃 =𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
𝑐𝑜𝑡𝜃 =𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃
Sum and Difference Identities
sin(𝛼 ± 𝛽) =𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 ± 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽
cos(𝛼 ± 𝛽) = 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 ∓ 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽
𝑡𝑎𝑛(𝛼 ± 𝛽) =𝑡𝑎𝑛𝛼 ± 𝑡𝑎𝑛𝛽1 ∓ 𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
Double and Half Angle Formulas
𝑠𝑖𝑛2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
𝑐𝑜𝑠2𝜃 = cos2 𝜃 − sin2 𝜃
𝑐𝑜𝑠2𝜃 = 2 cos2 𝜃 − 1
𝑐𝑜𝑠2𝜃 = 1 − 2 sin2 𝜃
𝑡𝑎𝑛2𝜃 =2𝑡𝑎𝑛𝜃
1 − tan2 𝜃
𝑠𝑖𝑛𝜃2= ±√
1 − 𝑐𝑜𝑠𝜃2
𝑐𝑜𝑠𝜃2= ±√
1 + 𝑐𝑜𝑠𝜃2
𝑡𝑎𝑛𝜃2= ±√
1 − 𝑐𝑜𝑠𝜃1 + 𝑐𝑜𝑠𝜃
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GSE Precalculus Trig Identities Review Name: _______________________ Block: __
Given that and are in quadrant 4 and = − =4 15
sin and cos5 17
, find:
1. cos(𝛼) 2. sin(𝛽) 3. sin(𝛼 + 𝛽) 4. cos(𝛼 − 𝛽) 5. tan(𝛼 − 𝛽) 6. sin(2𝛼) 7. cos(2𝛽) 8. tan(2𝛽)
9. If 1sin3
= and 90 180 , then find the value of sec
Use sum/difference formulas to find the exact value of the following: 10. = − sin60 sin(90 30 )
11. = − cos75 cos(120 45 ) Write as the sin, cos, or tan of a single angle. 12. 𝑠𝑖𝑛70𝑜𝑐𝑜𝑠40𝑜 − 𝑐𝑜𝑠70𝑜𝑠𝑖𝑛40𝑜 13. 𝑐𝑜𝑠210𝑜𝑐𝑜𝑠80𝑜 + 𝑠𝑖𝑛210𝑜𝑠𝑖𝑛80𝑜
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Verify the following.
=1.sec cot csc − =2 22.sin csc sin cos
+ + − =3.sin( ) sin( ) 2sin cosx y x y x y + =
csc cos4. 2cot
sec sin
=2sec
5. sec csctan
+ =2 26.cos (1 tan ) 1x x
+ =2 2 27.1 sec sin sec − =− +
1 18. 2csc cot
1 cos 1 cosx x
x x
+ + +=
−−
2
2
sin 5sin 6 sin 39.
sin 2sin 4
10. 𝑠𝑖𝑛 𝑥 (1 − 2 𝑐𝑜𝑠2 𝑥 + 𝑐𝑜𝑠4 𝑥) = 𝑠𝑖𝑛5 𝑥