1. 13.1 Trigonometric Identities
Objective: Use trigonometric identities to
find trigonometric values

2. What is a mathematical identity?
An equation that is true no matter what values replace the variables. Examples:
Identities/Properties we have used this semester!

3. Determine the six trigonometric ratios for :
sin  = csc  =
cos  = sec  =
tan  = cot  = y x
𝒚 𝒙
𝟏 𝒚
𝟏 𝒙
𝒙 𝒚

4. Reciprocal Identities
sin  = csc 
cos  sec 
tan  cot  y

5. Quotient Identities tan  cot 
*Also, since cot  is the reciprocal function of tan,
if , then

6. Pythagorean Identities

7. These are the identities you are responsible for!
On page 873 in your textbook
We will not be discussing Cofunction identities
and Negative Angle identities

8. Example 1: sin2 + cos2 = 1 Trigonometric identity Subtract.
Objective: Use Trigonometric Identities to Find Trigonometric Values
Example 1:
sin2 + cos2 = 1 Trigonometric identity
Subtract.
Take the square root of each side.
Answer: Since  is in the second quadrant, sin  is positive. Thus,

9. Example 2: Find cot  if sec  = –2 and 180< < 270.
Objective: Use Trigonometric Identities to Find Trigonometric Values
Example 2: Find cot  if sec  = –2 and 180< < 270.
tan2 + 1 = sec2 Trigonometric identity
tan2 = sec2 – 1 Subtract 1 from each side.
tan2 = (–2)2 – 1 Substitute –2 for sec .
tan2 = 4 – 1 Square –2.
tan2 = 3 Subtract.
Take the square root of each side.
Don’t forget -> All Students Take Calculus
cot  = 1 ± 3 = ± 3 3
Reciprocal identity
Answer: Since  is in the third quadrant, tan  is positive. Thus, cot  is positive and cot  =

10. Your turn! B. Find sin  if cot  = 2 and 180< < 270. A. B.
− 5 A. B. C. D. −
5 5 5

11. Your turn! Find tan  if cos  = 𝟏 𝟑 and 270< < 360.
Find csc  if sin  = − 𝟏 𝟐 and 180< < 270.

12. Homework
p. 876 #9, 12, 15, 16, 17, 19, 20