Download 11.3 Fundamental Trigonometric Identities

11.3 Fundamental Trigonometric Identities
Objectives:
F.TF.8: Prove the Pythagorean identity sin2 θ + cos2 θ = 1 and use it to find sin θ, cos θ, or tan θ
and the quadrant of the angle.
For the Board: You will be able to use fundamental trigonometric identities to simplify and rewrite
expressions and to verify other identities.
Bell Work 11.3:
Simplify:
 sin A  cos2 A 


1. 
 cos A  sin A 
 sin A  1 


2. tan A
 tan A  sin A 
Anticipatory Set:
x 2 y2
Recall: x + y = r on the unit circle. Therefore: 2  2  1
r
r
2
2
x  y
Applying an exponent rule:       1
r r
Recall: cos θ = x/r and sin θ = y/r so the above becomes cos2 θ + sin2 θ = 1
2
2
2
This is known as the Pythagorean Identity.
Fundamental Trigonometric Identities
Reciprocal Identities
1
csc θ =
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
Tangent and Cotangent Ratio Identities
sin θ
cos θ
tan θ =
cot θ =
cos θ
sin θ
Pythagorean Identities
cos2 θ + sin2 θ = 1
1 + tan2 θ = sec2 θ
cot2 θ + 1 = csc2 θ
Negative-Angle Identities
sin(-θ) = - sin θ
cos(-θ) = cos θ
tan(-θ) = - tan θ
Open the book to page 772 and read example 1.
Example: Prove each trigonomet4ric identity.
sec θ
a. tan θ =
csc θ
=
1
cos θ  1  1
1
cos θ sin θ
sin θ
1 sin θ sin θ



cos θ 1
cos θ
 tan θ
b. 1 – cot θ = 1 + cot(-θ)
1
1
 1
=1 
tan (θ)
- tan θ
1
1
 1  cot θ
tan θ
White Board Activity:
Practice: Prove each trigonometric identity.
a. sin θ cot θ = cos θ
b. 1 – sec(-θ) = 1 – sec θ
cos θ
sin θ ·
= cos θ
1 – sec(-θ) = 1 – (1/cos (-θ)) = 1 – (1/cos θ) = 1 – sec θ
sin θ
Open the book to page 773 and read example 2.
Example: Rewrite each expression in terms of cos θ, and simplify.
a. sec θ (1 – sin2 θ)
b. sin θ cos θ (tan θ + cot θ)
(1/cos θ)(1 – (1 – cos2 θ)
sin θ cos θ tan θ + sin θ cos θ cot θ
sin θ
cos θ
(1/cos θ)(1 – 1 + cos2 θ)
sin θ cos θ
 sin θ cos θ
cos θ
sin θ
2
2
2
(1/cos θ)(cos θ)
sin θ + cos θ
cos θ
1 – cos2 θ + cos2 θ
1 – 2 cos2 θ
White Board Activity:
Practice: Rewrite each expression in terms of sin, and simplify.
a. cos2/(1 – sin2)
b. cot2
Assessment:
Question Student pairs.
Independent Practice:
Text: pg. 775 prob. 1 – 6, 8 – 15.
Practice: pg. 775 prob. 17 – 37.