Basic Trigonometric Identities and Equations PreCalculus Teacher Mrs

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Basic Trigonometric Identities and Equations Pre-Calculus Teacher – Mrs. Volynskaya

Basic Trigonometric Identities and Equations Pre-Calculus Teacher – Mrs. Volynskaya

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

Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin 2 q + cos 2 q = 1 tan 2 q + 1 = sec 2 q cot 2 q + 1 = csc 2 q sin 2 q = 1 - cos 2 q tan 2 q = sec 2 q - 1 cot 2 q = csc 2 q - 1 cos 2 q = 1 - sin 2 q 5. 4. 3

Where did our pythagorean identities come from? ? Do you remember the Unit Circle?

Where did our pythagorean identities come from? ? Do you remember 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

Using the identities you now know, find the trig value. 1. ) If cosθ

Using the identities you now know, find the trig value. 1. ) If cosθ = 3/4, find secθ 2. ) If cosθ = 3/5, find cscθ.

3. ) sinθ = -1/3, find tanθ 4. ) secθ = -7/5, find sinθ

3. ) sinθ = -1/3, find tanθ 4. ) secθ = -7/5, find sinθ

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) 5. 4. 5

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.

Simplifying trig Identity Example 1: simplify tanxcosx sin x tanx cos x tanxcosx =

Simplifying trig Identity Example 1: simplify tanxcosx sin x tanx cos x tanxcosx = sin x

Simplifying trig Identity Example 2: simplify sec x csc x 1 cos sec x

Simplifying trig Identity Example 2: 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

Simplifying trig Identity Example 2: simplify cos 2 x - sin 2 x cos

Simplifying trig Identity Example 2: simplify cos 2 x - sin 2 x cos 2 x - sin 12 x cos x = sec x

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

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

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

Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

Practice 1 cos 2θ cosθ sin 2θ cos 2θ secθ-cosθ csc 2θ cotθ tan

Practice 1 cos 2θ cosθ sin 2θ cos 2θ secθ-cosθ csc 2θ cotθ tan 2θ

One way to use identities is to simplify expressions involving trigonometric functions. Often a

One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this: substitute using each identity simplify

Another way to use identities is to write one function in terms of another

Another way to use identities is to write one function in terms of another function. Let’s see an example of this: This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

(E) Examples • Prove tan(x) cos(x) = sin(x) 20

(E) Examples • Prove tan(x) cos(x) = sin(x) 20

(E) Examples • Prove tan 2(x) = sin 2(x) cos-2(x) 21

(E) Examples • Prove tan 2(x) = sin 2(x) cos-2(x) 21

(E) Examples • Prove 22

(E) Examples • Prove 22

(E) Examples • Prove 23

(E) Examples • Prove 23