Using Fundamental Identities

MATH 109 - PrecalculusS. Rook

Overview

• Section 5.1 in the textbook:– Using identities to evaluate a function– Simplifying trigonometric expressions– Factoring trigonometric expressions– Adding & subtracting trigonometric expressions– Eliminating fractions in a trigonometric expression

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Using Identities to Evaluate a Function

Review of Identities Discussed thus Far

• Examine the chart on page 374 – you must be familiar with these identities:– Reciprocal– Quotient/Ratio– Pythagorean– Cofunction– Even/Odd

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Using Identities to Evaluate a Function

• Given the value of a trigonometric function, sometimes we wish to deduce the value of another trigonometric function:– Try to get the values in terms of sines & cosines

5

Using Identities to Evaluate a Function (Example)

Ex 1: Use the given values to evaluate (if possible) all six trigonometric functions:

a) ,

b)

6

5

12cot x

12

13sec x

0sin ,2tan

Simplifying Trigonometric Expressions

Simplifying Trigonometric Expressions

• There are times when we wish to write a trigonometric expression in an alternate form:– Identities are the key

• As expected, there is usually more than one way to simplify a trigonometric expression

• Some tips when simplifying:– Check to see if anything can be factored out– Express trigonometric functions in terms of sines

& cosines8

Simplifying Trigonometric Expressions (Example)

Ex 2: Simplify using the fundamental identities – possible to have multiple correct answers:

a)

b)

9

xxx 222 sintantan

tan

sinsec

Simplifying Trigonometric Expressions (Example)

Ex 3: Use the trigonometric substitution to write the algebraic expression as a trigonometric function of θ, 0 < θ < π⁄2

10

cos3 ,9 2 xx

Factoring Trigonometric Expressions

Factoring Trigonometric Expressions

• Recall our strategies for factoring a quadratic:– i.e. easy & hard trinomials, difference of two

squares, etc• Can apply the same strategies when trigonometric

functions are involved:– E.g. – Permissible to substitute

• Just remember to revert back to the trigonometric function at the end

• E.g. Let x = csc θ:

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2csc3csc23652 xxxx

6csc5csc2

Factoring Trigonometric Expressions (Continued)

• We must have the SAME trigonometric function in an expression In order to factor– Pythagorean identities are helpful in this situation– e.g. cannot be factored initially

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2tansec2

Factoring Trigonometric Expressions (Example)

Ex 4: Factor the trigonometric expression and then simplify if possible:

a)

b)

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1csccsccsc 23 xxx

11sincos2 xx

Adding & Subtracting Trigonometric Expressions

Adding & Subtracting Trigonometric Expressions

• Sometimes we wish to condense multiple sums & differences of fractions containing trigonometric functions into one fraction– Find an LCD• e.g. LCD of is (sin x)(cos x) • e.g. LCD of is (sec x)(sec x + 1)

• e.g. LCD of is tan2θ

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xx cos

2

sin

4

1sec

1

sec

1

xx

tan

3

tan

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Adding & Subtracting Trigonometric Expressions (Example)

Ex 5: Simplify by combining over one fraction:

a)

b)

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costansin

xx

cotsin

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Eliminating Fractions in a Trigonometric Expression

Eliminating Fractions in a Trigonometric Expression

• An important skill to master especially for Calculus is to be able to rewrite a trigonometric expression without a fraction

• The main idea is to get ONE term in the denominator– We can then divide all terms of the numerator by the

denominator• Only works when there is ONE term in the denominator

• If the denominator contains two terms:– Multiply by the conjugate of the denominator

• Results in a difference of two squares– Apply a Pythagorean identity to condense to one term

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Eliminating Fractions in a Trigonometric Expression (Example)

Ex 6: Rewrite the expression so that it is not in fractional form:

a)

b)

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y

y

cos1

sin 2

xx tansec

3

Summary

• After studying these slides, you should be able to:– Utilize the identities on page 474– Use identities to evaluate trigonometric functions– Simplify trigonometric expressions– Factor trigonometric expressions– Add & subtract trigonometric expressions– Eliminate fractions in a trigonometric expression

• Additional Practice– See the list of suggested problems for 5.1

• Next lesson– Verifying Trigonometric Identities (Section 5.2)

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