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

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

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