Basic Trigonometric Identities Pythagorean Identity Trigonometric Ratios Consider

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Basic Trigonometric Identities Pythagorean Identity

Basic Trigonometric Identities Pythagorean Identity

Trigonometric Ratios • Consider a right angle triangle with as one of its acute

Trigonometric Ratios • Consider a right angle triangle with as one of its acute angles. The trigonometric ratios are defined as follows (see Figure 1). Figure 1

Unit Circle

Unit Circle

A circle with center at (0, 0) and radius 1 is called a unit

A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (0, 1) (-1, 0) (0, -1) So points on this circle must satisfy this equation.

Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin 2 + cos 2 =

Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin 2 + cos 2 = 1 tan 2 + 1 = sec 2 cot 2 + 1 = csc 2 sin 2 = 1 - cos 2 tan 2 = sec 2 - 1 cot 2 = csc 2 - 1 cos 2 = 1 - sin 2

Where did our pythagorean identities come from? ? …. and the Unit Circle? •

Where did our pythagorean identities come from? ? …. and the Unit Circle? • What is the equation for the unit circle? x 2 + y 2 = 1 • What does x = ? What does y = ? (in terms of trig functions) sin 2θ + cos 2θ = 1 Pythagorean Identity!

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos 2θ sin 2θ + cos 2θ = 1. cos 2θ tan 2θ + 1 = sec 2θ Quotient Identity another Pythagorean Identity Reciprocal Identity

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin 2θ + cos 2θ = 1. sin 2θ 1 + cot 2θ = csc 2θ Quotient Identity a third Pythagorean Identity Reciprocal Identity

Simplifying trig Identity Example: Simplify tanx cosx sin x tanx cos x tanxcosx =

Simplifying trig Identity Example: Simplify tanx cosx sin x tanx cos x tanxcosx = sin x

Simplifying trig Identity Example: simplify sec x csc x 1 cos sec x csc

Simplifying trig Identity Example: simplify sec x csc x 1 cos sec x csc 1 x sin x = 1 sinx x cos x 1 = sin x cos x = tan x

Using the identities you now know, find the trig value. If cosθ = 3/4,

Using the identities you now know, find the trig value. If cosθ = 3/4, find secθ If cosθ = 3/5, find cscθ.

Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. a) b)

Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. a) b)

Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x

Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d)

Simplify each expression

Simplify each expression

Example Simplify: = cot x (csc 2 x - 1) = cot x (cot

Example Simplify: = cot x (csc 2 x - 1) = cot x (cot 2 x) = cot 3 x Factor out cot x

Example Simplify: = sin x (sin x) + cos x 2 = sin x

Example Simplify: = sin x (sin x) + cos x 2 = sin x + (cos x)cos x = sin 2 x + cos 2 x cos x = 1 cos x = sec x Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean iden Use reciprocal identity

Examples • Prove tan(x) cos(x) = sin(x) 21

Examples • Prove tan(x) cos(x) = sin(x) 21

Examples • Prove tan 2(x) = sin 2(x) cos-2(x) 22

Examples • Prove tan 2(x) = sin 2(x) cos-2(x) 22

Examples • Prove 23

Examples • Prove 23

Examples • Prove 24

Examples • Prove 24