Trig Identities-Trig Equation

8/7/2019 Trig Identities-Trig Equation

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TRIG IDENTITIES

What are they used for?To simplify trig expressions

To solve trig equations

What is the idea behind Trig Identities?

To establish relations between the side

measures of similar right triangles.


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THE DIFFERENCE

An Equation is an Algebraic expression

An Identity is a rule where all values ofthe domain common to the trig functions

involved are solutions to this equation.


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BASIC IDENITITIES


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IDENTITY #1

sin


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IDENTITY #2

1 + tan


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IDENTITY #3

1 + cot


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THE THREE IDENTITIES

1) sin2 + cos2 = 1N.B. sin2 = (sin )2 . That could be confused with sin2

2) 1 + tan2 = sec2

N.B. sec2 = 1/cos2

3) 1+ cot2 = csc2 N.B. cot2 = 1/tan2 ; csc2 = 1/sin2


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PROVING YOUR IDENTITY ALGEBRAIC

METHOD

This method involves; referring to definitions of trig functions,basic identities, factorization, number sense and operations.

1. To prove an identity, identify the most complex sideand simplify it to express it in the same terms as the

other side. To do so, try the following:2. Substitute on or more basic identities to simplify the

expression.

3. Perform operations or factor to find a basic identity or

a factor common to the numerator and denominator.4. Multiply the numerator and the denominator by the

same trigonometric expression.

5. Express the various functions using sine and cosine

functions.


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EXAMPLE OF PROOF:

It helps to look at a Left Hand Side & a Right Hand Side

1

= 1Quad EratDemonstratum


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EXAMPLE OF PROOF:


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EXAMPLE OF PROOF:


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EXAM QUESTION

Prove that,

v

x

x

x

x

xx 2

2

2

2

22

t nc s

c s1

1s c

1t nsin!

v

Show your work.


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EXAM QUESTION

For all values of A (for which A is defined), the expressiontan A + cot A is equal to

A) sin A cos A. C) sec A cosec A.

B) sec A cos A. D) sin A cosec A.


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SOLVING TRIG EQUATIONS


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ANOTHER EXAMPLE


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ONE MORE EXAMPLE


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EXAMPLE CONTINUED...

By breaking it down into two factors:(2 cos


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STEPS TO HELP SOLVE TRIG EQUATIONS

1. State the restrictions that apply to the functions

associated with these forms of equations.

2. Use identities to transform the equation so that it

contains a single trig function or become a familiartype of equation.

3. Find the solutions for the equation over [0, p[ where p

is the period of the function

4. Verify the solution obtained

5. Find the general solution by considering the period of

the function.


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HOMEWORK

Page 291

#1, 2, 3, 5, 8ab, 11


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MORE PRACTICE ON TRIG IDENTITY

Page 305

# 49, 52, 53, 54, 58

Trig Identities-Trig Equation

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