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

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!

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

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

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

Pythagorean Identities

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

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,

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  = 3 3

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

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

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