BELL-WORK TCAP Bell-Work # What is the cotangent of angle x if sec(x) = 12 5

HW 3.3(d) Due Monday: Handout # 9,11,13,15

HW 3.3(c) Solutions √ √53 7

HW 3.3(c) Solutions 7.√3 8.√3 9.2√ √ tan 3 (x) ORcot(x) tan(x) 13.sin(x) cos(x)

HW 3.3(c) Solutions 14.cos(x) sin(x) sin 3 (x)cos(x) 16.√ √ √ √

HW 3.3(c) Solutions 21.√ √3 3

Guiding question: What is a trigonometric identity?

Right Triangle Trigonometry Now that we know all the trig ratios, we will investigate some other trigonometric properties.

Mathematical Identities What is an identity? An identity is an equation that is true for any value of the variable. We have already discussed the reciprocal identities: sin x = 1. csc x cos x = 1. sec x tan x = 1. cot x

Tangent Identity tan x = sin x cos x

Tangent Identity Proof: tan x = x y sin x = x z cos x = y z sin x = cos x x ÷ y z z = x y

Cotangent Identity In the same manor, cot x = cos x sin x

Trigonometric Identities Represent each of the following in terms of sin(x) and cos(x). You may assume that each angle x is measured in degrees. sec(x)∙tan(x) cot(x)∙sin(x) cos 2 (x)∙tan(x) csc(x) cot(x)

Pythagorean Identities Derived from the Pythagorean theorem… sin 2 x + cos 2 x = 1

Pythagorean Identities Proof: By Pythagoras… x 2 + y 2 = z 2 Dividing through by z 2 … x 2 + y 2 = 1 z 2 z 2 Since sin θ = x&cos θ = y z sin 2 θ = x 2 &cos 2 θ = y 2 z 2 z 2 sin 2 θ + cos 2 θ = 1

Pythagorean Identities Intuitively then, cos 2 θ = 1 – sin 2 θ sin 2 θ = 1 – cos 2 θ

Pythagorean Identities tan 2 θ + 1 = sec 2 θ Proof: We know sin 2 θ + cos 2 θ = 1. Dividing by cos 2 θ… sin 2 θ + cos 2 θ = 1. cos 2 θ cos 2 θ cos 2 θ tan 2 θ + 1 = sec 2 θ

Pythagorean Identities 1 + cot 2 θ = csc 2 θ Proof: We know sin 2 θ + cos 2 θ = 1. Dividing by sin 2 θ… sin 2 θ + cos 2 θ = 1. sin 2 θ sin 2 θ sin 2 θ 1 + cot 2 θ = csc 2 θ

Trigonometric Identities We can use these identities to do further proofs! Show that sec 2 θ + csc 2 θ = sec 2 θ csc 2 θ cos 2 x sin 2 x Common denominator… sin 2 x + cos 2 x cos 2 x sin 2 x 1. cos 2 x sin 2 x sec 2 θ csc 2 θ

Trigonometric Identities Handout

Who wants to answer the Guiding question? What is a trigonometric identity?