What this module is about

This module is about trigonometric equations and proving fundamental identities. The lessons in this module were presented in a very simple way so it will be easy for you to understand solve problems without difficulty. Your knowledge in previous lessons would be of help in the process

What you are expected to learn

This module is designed for you to: 1. state the fundamental identities 2. prove trigonometric identities 3. state and illustrate the sum and cosine formulas of cosine and sine 4. determine the sine and cosine of an angle using the sum and difference formulas. 5. solve simple trigonometric equations

How much do you know

A. Answer the following: 1. Which of the following does not equal to 1 for all A in each domain? a. sin2 A + cos2 A c.

b. sec2 A - cos2 A d. tan A cot A

sin A sec A

2. Simplify cos2 A sec A csc A

3. If sin ∝ =

12 4 and cos β = , where ∝ and β are both in the first 13 5 quadrant, find the values of cos (∝ + β ). 4. Sec A is equal to a. cos A b. sin A c.

1 cos A

d.

1 . sin A

5. Express

1 − csc B in terms of cos B and Sin A. cot B 1 − sin B a. cos B – sinB b. c. sin B – cos B cos B cos φ . sin φ cot φ b. tanφ c. –csc φ d. -1

d.

sin B − 1 cos B

6. Simplify a. 1

7. Multiply and simplify ( 1 – cos2 t ) ( 1 + tan2 t ). 8. Express tan B ( sin B + cot B + cos B ) in terms of sec B. 9. Compute sin

5π π π from the function of and . 12 4 6

10. Solve the equation cos A – 2sin A cos A = 0.

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What you will do

Lesson 1 Fundamental Trigonometric Identities

To be able to simplify trigonometric expressions and solve trigonometric equations, you must be able to know the fundamental trigonometric identities. The Eight Fundamental Identities: A. Reciprocal Relations 1. sec θ =

1 cosθ 1 sin θ 1 tan θ

2. csc θ =

3. cot θ =

B. Quotient Relations 4. tan θ =

sin θ cosθ cosθ sin θ

5. cot θ =

C. Pythagorean Relations 6. cos2 θ + sin2 θ = 1 7. 1 + tan2 θ = sec2 θ 8. cot2 θ + 1 = csc2 θ

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With the aid of this identities, you may now simplify trigonometric expressions. Examples: Perform the indicated operation. a. ( 1 – sin x ) ( 1 + sin x ) = 1 - sin2 x = cos2 x b. ( sec A – 1 ) ( sec A + 1 ) = sec2 A - 1 = tan2 A c. tan θ ( cot θ + tan θ ) = tan θ cot θ + tan2 θ = 1 + tan = sec2 θ tan θ cot θ = 1, because sin θ sin θ tan θ = cot θ = cosθ cosθ sin θ sin θ tan θ cot θ = ( )( ) = 1 cosθ cosθ d. cos x ( sec x - cos x ) = cos x sec x - cos2 x = 1 - cos2 x = sin2 x cos x sec x = 1, because sec x = therefore, cos x sec x = cos x ( Pythagorean Relation no. 2 Product of sum & difference of 2 terms Since, cos2 θ + sin2 θ = 1, then 1 - sin2 x = cos2 x

1 cos x

1 ) = 1 cos x

4

sin 2 B e. cos B + cos B

=

cos 2 B + sin 2 B cos B 1 cos B

sec B

cos B, Least common denominator

= =

Identity C. 6 Identity A. 1

Simplify the following expressions to a single function. a. cos 2 A tan2 A = sin2 A cos 2 A (

sin 2 A ) = sin2 A 2 cos A

b. ( sin x + cos x )2 + ( sin x - cos x )2 = sin2 x + 2sinx cos x + cos2x + sin2x - 2 sin x cos x + cos2x = = sin2 x + cos2x + sin2x + cos2x 2

since, cos2 θ + sin2 θ = 1 c. cot B sec B sin B = 1 since, cot θ =

cosθ and sec θ = sin θ

1 cosθ

then, (

cosθ 1 )( ) sin θ = 1 sin θ cosθ

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d. csc A - csc A cos2 A = sin A = csc A ( 1 - cos2 A ) = = = csc A ( sin2 A ) Factor csc A Identity C. 6 Identity A. 2 Cancellation

1 ( sin2 A) sin θ

sin A

e. cos3 B + cos B sin2 B = cos B ( cos2 B + sin2 B ) = cos B ( 1 ) = cos B You are now ready to prove identities. In this lesson, you will prove that one side of the equation is equal to the other side. You can work on either of the two sides to verify the expressions are equal or you can work on both equations to arrive at an equal statement. Suggested Steps in Proving Identities 1. Start with the more complicated side and transform it into the simpler side. 2....