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. 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.

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7.1 – Basic Trigonometric Identities and Equations

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7 1 basic trigonometric identities and equations

7.1 – Basic Trigonometric Identities and Equations


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


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?

x2 + y2 = 1

  • What does x = ? What does y = ?

  • (in terms of trig functions)

sin2θ + cos2θ = 1

Pythagorean Identity!


Take the pythagorean identity and discover a new one

Take the Pythagorean Identity and discover a new one!

Hint: Try dividing everything by cos2θ

sin2θ + cos2θ = 1 .

cos2θcos2θ cos2θ

tan2θ + 1 = sec2θ

Quotient

Identity

Reciprocal

Identity

another Pythagorean Identity


Take the pythagorean identity and discover a new one1

Take the Pythagorean Identity and discover a new one!

Hint: Try dividing everything by sin2θ

sin2θ + cos2θ = 1 .

sin2θsin2θ sin2θ

1 + cot2θ = csc2θ

Quotient

Identity

Reciprocal

Identity

a third Pythagorean Identity


Using the identities you now know find the trig value

Using the identities you now know, find the trig value.

1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.


7 1 basic trigonometric identities and equations

3.) sinθ = -1/3, find tanθ

4.) secθ = -7/5,find sinθ


7 1 basic trigonometric identities and equations

Simplifying Trigonometric Expressions

Identities can be used to simplify trigonometric expressions.

Simplify.

b)

a)

5.4.5


7 1 basic trigonometric identities and equations

Simplifing Trigonometric Expressions

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

d)


Simplify each expression

Simplify each expression.


Simplifying trig identity

sin x

cos x

Simplifying trig Identity

Example1: simplify tanxcosx

tanx cosx

tanxcosx = sin x


Simplifying trig identity1

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

Simplifying trig Identity

Example2: simplify


Simplifying trig identity2

cos2x - sin2x

cos2x - sin2x

cos2x - sin2x

1

= sec x

cos x

cos x

Simplifying trig Identity

Example2: simplify


Example

Example

Simplify:

Factor out cot x

= cot x (csc2 x - 1)

Use pythagorean identity

= cot x (cot2 x)

Simplify

= cot3 x


Example1

= 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

Example

Simplify:

Use quotient identity

Simplify fraction with LCD

Simplify numerator

Use pythagorean identity

Use reciprocal identity

= sec x


Your turn

Your Turn!

Combine fraction

Simplify the numerator

Use pythagorean identity

Use Reciprocal Identity


Practice

Practice


7 1 basic trigonometric identities and equations

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


7 1 basic trigonometric identities and equations

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

(E) Examples

  • Prove tan(x) cos(x) = sin(x)


E examples1

(E) Examples

  • Prove tan2(x) = sin2(x) cos-2(x)


E examples2

(E) Examples

  • Prove


E examples3

(E) Examples

  • Prove


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