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7.1 – Basic Trigonometric Identities and Equations

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7.1 – Basic Trigonometric Identities and Equations

Trigonometric Identities

Quotient Identities

Reciprocal Identities

Pythagorean Identities

sin2q + cos2q = 1

tan2q + 1 = sec2q

cot2q + 1 = csc2q

sin2q = 1 - cos2q

tan2q = sec2q - 1

cot2q = csc2q - 1

cos2q = 1 - sin2q

5.4.3

Where did our pythagorean identities come from??

- What is the equation for the unit circle?

x2 + y2 = 1

- What does x = ? What does y = ?
- (in terms of trig functions)

sin2θ + cos2θ = 1

Pythagorean Identity!

Hint: Try dividing everything by cos2θ

sin2θ + cos2θ = 1 .

cos2θcos2θ cos2θ

tan2θ + 1 = sec2θ

Quotient

Identity

Reciprocal

Identity

another Pythagorean Identity

Hint: Try dividing everything by sin2θ

sin2θ + cos2θ = 1 .

sin2θsin2θ sin2θ

1 + cot2θ = csc2θ

Quotient

Identity

Reciprocal

Identity

a third Pythagorean Identity

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

Simplifying Trigonometric Expressions

Identities can be used to simplify trigonometric expressions.

Simplify.

b)

a)

5.4.5

Simplifing Trigonometric Expressions

c) (1 + tan x)2 - 2 sin x sec x

d)

sin x

cos x

Example1: simplify tanxcosx

tanx cosx

tanxcosx = sin x

sec x

sec x

sin x

sinx

1

1

1

= tan x

=

=

x

cos x

cos x

cos x

csc x

csc x

sin x

1

Example2: simplify

cos2x - sin2x

cos2x - sin2x

cos2x - sin2x

1

= sec x

cos x

cos x

Example2: simplify

Simplify:

Factor out cot x

= cot x (csc2 x - 1)

Use pythagorean identity

= cot x (cot2 x)

Simplify

= cot3 x

= 1

= sin2 x + cos2x

= sin x (sin x) + cos x

= sin2 x + (cos x)

cos x

cos x

cos x

cos x

cos x

cos x

Simplify:

Use quotient identity

Simplify fraction with LCD

Simplify numerator

Use pythagorean identity

Use reciprocal identity

= sec x

Combine fraction

Simplify the numerator

Use pythagorean identity

Use Reciprocal Identity

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 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.

- Prove tan(x) cos(x) = sin(x)

- Prove tan2(x) = sin2(x) cos-2(x)

- Prove

- Prove