Chapter 5.1 Fundamental Identities

1. Chapter 5.1Fundamental Identities Learning Target: I can use the fundamental identities to simplify trigonometric expressions and solve trigonometric equations.

2. Basic Trig Identities • Csc (θ) = 1/sin(θ) • Sec (θ) = 1/cos(θ) • Cot(θ) = 1/tan(θ) • Sin(θ) = 1/csc(θ) • Cos(θ) = 1/sec(θ) • Tan(θ) = 1/cot(θ) • Tan(θ) = sin(θ)/cos(θ) • Cot(θ) = cos(θ)/sin(θ)

3. Pythagorean Identities • (sin t)2 + (cos t)2 = 1 • (cos t)2/(cos t)2 + (sin t)2/(cos t)2 = 1/(cos t)2 → 1 + (tan t)2 = (sec t)2 (cos t)2/(sin t)2 + (sin t)2/(sin t)2 = 1/(sin t)2 → (cot t)2 + 1 = (csc t)2

4. Cofunction Identities

5. Odd-Even Identities sin (–x) = –sin x cos (–x) = cos x tan (–x) = –tan x csc (–x) = –csc x sec (–x) = sec x cot (–x) = –cot x

6. Example • Simplify by expanding and using identities. • (sec(x) + 1)(sec(x) - 1)/(sin2(x))

7. Example • Simplify by combining fractions and using identities. • Cos(x)/(1-sin(x)) – sin(x)/cos(x)

8. Example • Find all values of x in the interval [0,2П] that solve cos3(x)/sin(x) = cot(x)

9. Homework Pg. 451 # 3 – 54 every 3rd