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Published byJacob Floyd Modified over 6 years ago

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**7.1 – Basic Trigonometric Identities and Equations**

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**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

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**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!

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**Take the Pythagorean Identity and discover a new one!**

Hint: Try dividing everything by cos2θ sin2θ + cos2θ = cos2θ cos2θ cos2θ tan2θ = sec2θ Quotient Identity Reciprocal Identity another Pythagorean Identity

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**Take the Pythagorean Identity and discover a new one!**

Hint: Try dividing everything by sin2θ sin2θ + cos2θ = sin2θ sin2θ sin2θ cot2θ = csc2θ Quotient Identity Reciprocal Identity a third Pythagorean Identity

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

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

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

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**Simplifying Trigonometric Expressions**

Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5

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**Simplifing Trigonometric Expressions**

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

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**Simplify each expression.**

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**Simplifying trig Identity**

Example1: simplify tanxcosx sin x cos x tanx cosx tanxcosx = sin x

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**Simplifying trig Identity**

sec x csc x Example2: simplify 1 cos x 1 cos x sinx = x sec x csc x 1 sin x = sin x cos x = tan x

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**Simplifying trig Identity**

cos2x - sin2x cos x Example2: simplify = sec x cos2x - sin2x cos x cos2x - sin2x 1

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**Example Simplify: = cot x (csc2 x - 1) Factor out cot x**

= cot x (cot2 x) Use pythagorean identity = cot3 x Simplify

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**Example Simplify: = sin x (sin x) + cos x Use quotient identity cos x**

Simplify fraction with LCD = sin2 x + (cos x) cos x = sin2 x + cos2x cos x Simplify numerator = cos x Use pythagorean identity = sec x Use reciprocal identity

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**Your Turn! Combine fraction Simplify the numerator**

Use pythagorean identity Use Reciprocal Identity

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Practice 1 cos2θ cosθ sin2θ cos2θ secθ-cosθ csc2θ cotθ tan2θ

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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

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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.

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

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

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(E) Examples Prove

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(E) Examples Prove

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