Precalculus 5.1 notes: Using Fundamental Identities

5.1 Using Fundamental Identities

The Identities can be found on the following web site:
http://www.sosmath.com/trig/Trig5/trig5/trig5.html
Reciprocal identities
Pythagorean Identities
Quotient Identities
Co-Function Identities
Even-Odd Identities

Example 1: Evaluate the trigonometric functions given
Tan x = 7/24 and sec x is negative

Tan x is positive in the I and III quadrant
Sec x is negative in the II and III quadrant
Therefore this can be evaluated by knowing we are in the III quadrant

7² + 24² = c²
49 + 576 = c²
625 = c²
25 = c

Therefore:
Sin x = -7/25
Csc x = -25/7
Cos x = -24/25
Sec x = -25/24
Tan x = 7/24
Cot x = 24/7

Example 2: Simplify
cot x sec x
((cos x)/ (sin x) ) (1/ cos x)
1/ sin x
Csc x

Example 3:

(Cos²y) / (1 – sin y)

(1 – sin²y)/ (1 – sin y)

(1 + sin y)(1 – sin y)/ (1 – sin y)

1 + sin y

Example 4:

(tan²θ)/ (sec² θ)

(sin² θ)/(cos² θ) ÷ (1 / (cos²)

(sin² θ)/(cos² θ) × (cos²)/1

sin² θ

Example 5: Verify

Cos θ sec θ – cos² θ = sin² θ

Cos θ (1/ cos θ) - cos² θ = sin² θ

1 – cos² θ = sin² θ

sin² θ = sin² θ

Example 6: verify

(sec² θ - tan² θ + tan θ)/ (sec θ) = cos θ + sin θ

(1 + tan² θ – tan² θ + tan θ)/(sec θ) = cos θ + sin θ

(1 + tan θ)/ sec θ = cos θ + sin θ

1/sec θ + tan θ / sec θ = cos θ + sin θ

cos θ + (sin θ / cos θ) / (1/ cos θ) = cos θ + sin θ

cos θ + sin θ = cos θ + sin θ

Example 7: Factor

sec²x tan²x + sec²

sec²x ( tan²x + 1)

sec²x (sec²x)

sec⁴x

5.1 Homework: #43; pg. 381; # 5, 7, 25, 27, 37 – 43 odd, 55 – 59 odd, 67 – 77 odd, 91, 93, 101, 103