14-1 Trigonometric Identities

Quick Review

A trigonometric identity is a trigonometric equation that is true for all values except those for which the expressions on either side of the equal sign are undefined.

Reciprocal Identities

• csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ co-secant theta equals . fraction 1 , over sine theta end fraction . secant theta equals . fraction 1 , over cosine theta end fraction . co-tangent theta equals . fraction 1 , over tangent theta end fraction

Tangent and Cotangent Identities

• tan θ = sin θ cos θ cot θ = cos θ sin θ tangent theta equals . fraction sine theta , over cosine theta end fraction . co-tangent theta equals . fraction cosine theta , over sine theta end fraction

Pythagorean Identities

cos 2 θ + sin 2 θ = 1 cosine squared , theta plus , sine squared , theta equals 1

1 + tan 2 θ = sec 2 θ 1 plus , tangent squared , theta equals , secant squared , theta

1 + cot 2 θ = csc 2 θ 1 plus , co-tangent squared , theta equals , co-secant squared , theta

Example

Simplify the trigonometric expression cot θ sec θ . co-tangent theta secant theta .

cot θ sec θ = cos θ sin θ · sec θ Cotangent Identity   = cos θ sin θ · 1 cos θ Reciprocal identity   = 1 sin θ Simplify .   = csc θ Reciprocal identity table with 4 rows and 3 columns , row1 column 1 , co-tangent theta secant theta , column 2 equals . fraction cosine theta , over sine theta end fraction . middle dot secant theta , column 3 cap cotangentcap identity , row2 column 1 , , column 2 equals . fraction cosine theta , over sine theta end fraction . middle dot . fraction 1 , over cosine theta end fraction , column 3 cap reciprocalidentity , row3 column 1 , , column 2 equals . fraction 1 , over sine theta end fraction , column 3 cap simplify , . , row4 column 1 , , column 2 equals co-secant theta , column 3 cap reciprocalidentity , end table

Exercises

Verify each identity. Give the domain of validity for each identity.

6. sin θ tan θ = 1 cos θ − cos θ sine theta tangent theta equals . fraction 1 , over cosine theta end fraction . minus cosine theta
7. cos 2 θ cot 2 θ = cot 2 θ − cos 2 θ cosine squared , theta , co-tangent squared , theta equals , co-tangent squared , theta negative , cosine squared , theta

Simplify each trigonometric expression.

8. 1 − sin 2 θ 1 minus , sine squared , theta
9. cos θ sin θ cot θ fraction cosine theta , over sine theta co-tangent theta end fraction
10. csc 2 θ − cot 2 θ co-secant squared , theta negative , co-tangent squared , theta
11. cos 2 θ − 1 cosine squared , theta negative 1
12. sin θ cos θ tan θ fraction sine theta cosine theta , over tangent theta end fraction
13. sec θ sin θ cot θ secant theta sine theta co-tangent theta

14-2 Solving Trigonometric Equations Using Inverses

Quick Review

The function cos − 1 cosine super negative 1 end super x is the inverse of cos θ cosine theta with the restricted domain 0 ≤ θ ≤ π . 0 less than or equal to theta less than or equal to pi . The function sin − 1 sine super negative 1 end super x is the inverse of sin θ sine theta with the restricted domain − π 2 ≤ θ ≤ π 2 negative , pi over 2 , less than or equal to theta less than or equal to , pi over 2 and tan − 1 tangent super negative 1 end super x is the inverse of tan θ tangent theta with the restricted domain − π 2 ≤ θ ≤ π 2 . negative , pi over 2 , less than or equal to theta less than or equal to , pi over 2 . .

Example

Solve 2 cos θ sin θ − 3 cos θ = 0 2 cosine theta sine theta negative square root of 3 cosine theta equals 0 for θ theta with 0 ≤ θ < 2 π . 0 less than or equal to theta less than 2 pi .

2 cos θ sin θ − 3 cos θ = 0   cos θ ( 2 sin θ − 3 ) = 0 Factor . cos θ = 0   or   2 sin θ − 3 = 0 Zero-Product Property . cos θ = 0   sin θ = 3 2 Solve for cos θ  and sin θ . θ = π 2   or   3 π 2   θ = π 3 or   2 π 3 Use the unit circle . table with 5 rows and 2 columns , row1 column 1 , 2 cosine theta sine theta minus square root of 3 cosine theta equals 0 , column 2 , row2 column 1 , cosine theta . open . 2 sine theta minus square root of 3 . close . equals 0 , column 2 cap factor , . , row3 column 1 , cosine theta equals 0 or 2 sine theta minus square root of 3 equals 0 , column 2 cap zerominuscap productcap property . . , row4 column 1 , cosine theta equals 0 sine theta equals , fraction square root of 3 , over 2 end fraction , column 2 cap solvefor cosine theta , and sine theta . , row5 column 1 , theta equals , pi over 2 , or , fraction 3 pi , over 2 end fraction , theta equals , pi over 3 , or , fraction 2 pi , over 3 end fraction , column 2 cap usetheunitcircle . . , end table

Exercises

Use a unit circle and 30 ° − 60 ° − 90 ° 30 degrees , minus 60 degrees negative 90 degrees triangles to find the value in degrees of each expression.

14. sin − 1 ( − 3 2 ) sine super negative 1 end super . open , negative , fraction square root of 3 , over 2 end fraction , close
15. tan − 1 3 tangent super negative 1 end super . square root of 3
16. tan − 1 ( − 3 3 ) tangent super negative 1 end super . open , negative , fraction square root of 3 , over 3 end fraction , close
17. cos − 1 3 2 cosine super negative 1 end super . fraction square root of 3 , over 2 end fraction

Use a calculator and inverse functions to find the value in radians of each expression.

18. sin − 1 sine super negative 1 end super 0.33
19. tan − 1 ( − 2 ) tangent super negative 1 end super . open negative 2 close
20. cos − 1 ( − 0 . 64 ) cosine super negative 1 end super . open negative 0 . 64 close
21. cos − 1 0 . 98 cosine super negative 1 end super . 0 . 98

Solve each equation for 0 ≤ θ < 2 π . 0 less than or equal to theta less than 2 pi .

22. 2 cos θ = 1 2 cosine theta equals 1
23. 3 tan θ = 1 square root of 3 tangent theta equals 1
24. sin θ = sin 2 θ sine theta equals , sine squared , theta
25. sec θ = 2 secant theta equals 2

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