1. 5
Trigonometric Identities

2. Trigonometric Identities5
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities

3. Fundamental Identities
5.1
Fundamental Identities
Fundamental Identities ▪ Using the Fundamental Identities

4. Fundamental Identities
Reciprocal Identities
Quotient Identities

5. Fundamental Identities
Pythagorean Identities
Negative-Angle Identities

6. Note
In trigonometric identities, θ can be an angle in degrees, a real number, or a variable.

7. If and θ is in quadrant II, find each function value.
Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
If and θ is in quadrant II, find each function value.
Pythagorean identity
(a) sec θ
In quadrant II, sec θ is negative, so

8. Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)

9. Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
(b) cot(– θ)
Reciprocal identity
Negative-angle identity

10. Caution
To avoid a common error, when taking the square root, be sure to choose the sign based on the quadrant of θ and the function being evaluated.

11. Write cos x in terms of tan x.
Example 2
EXPRESSING ONE FUNCITON IN TERMS OF ANOTHER
Write cos x in terms of tan x.
Since sec x is related to both cos x and tan x by identities, start with
Take reciprocals.
Reciprocal identity
Take the square root of each side.
The sign depends on the quadrant of x.

12. Write in terms of sin θ and cos θ, and
Example 3
REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE
Write in terms of sin θ and cos θ, and
then simplify the expression so that no quotients appear.
Quotient identities
Multiply numerator and denominator by the LCD.

13. Example 3 REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE (cont’d)
Distributive property
Pythagorean identities
Reciprocal identity

14. Caution
When working with trigonometric expressions and identities, be sure to write the argument of the function.
For example, we would not write An argument such as θ is necessary in this identity.