Packet

Math 240 Class Packet #3

Topics

- Introduction to identities - Defining trig functions in terms of other trig functions - Deriving Pythagorean identities - Working with identities to evaluate other trig functions

First Things First

1) Complete the table given below by expressing each of the given trig functions in terms of a different trig function,

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2) If x2 + y2 = r2, use what you know about Sine and Cosine to derive the first Pythagorean identity, cos2 x + sin2 x = 1.

3) Use the result from (2) to derive the second Pythagorean identity, 1 + tan2 x = sec2 x.

4) For each of the following, use a Pythagorean identity to evaluate the given expression, then verify your result without using the Pythagorean identity.

a) Find sin θ if cos θ = 5 13

and θ terminates in Q1.

b) Find cos θ if sin θ = 1 3 and θ terminates in Q2.

5) If sin θ = 12 13

and θ terminates in Q1, find all other trigonometric ratios of θ.

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�� ��� θ =

��� θ = ��� θ =

��� θ = ��� θ =

Do it Yourself

6) If Sin θ = 2 3 , use the result of (1) to find Csc θ. Explain why we do not have enough information to find

Cos θ.

7) Use the first or second Pythagorean identity to derive the third identity 1 + cot2 θ = csc2 θ and summa- rize your results below.

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���� θ + ���� θ =

� + ���� θ =

� + ���� θ =

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8) For each of the following, use a Pythagorean identity to evaluate the given expression, then verify your result without using the Pythagorean identity.

a) Find cos θ if sin θ = -4 5

and θ terminates in Q4.

b) Find sin θ if cos θ = 1 5 and θ terminates in Q4.

9) If cos θ = -1 3

and θ terminates in Q2, find all other trigonometric ratios of θ.

��� θ = ��� θ =

��� θ = -�

� ��� θ =

��� θ = ��� θ =

Class Packet #3.nb ���3

Math 240 Class Packet #4

Topics

- Simplifying trig expression - Calculators - Proving basic trig identities

First Things First

1) Write each of the following in terms of sin θ and cos θ only.

a) sec θ tan θ csc θ

b) csc θ cot θ

c) tan θ + sec θ

2) Simplify each of the expressions given below (leave your answer in terms of sin θ or cos θ only).

a) 1 sin θ

- sin θ

b) (cos θ + 1) (cos θ - 4)

3) For each of the following, use your calculator to determine whether or not the given equation appears to be true. Copy down the window that you used to view the graphs.

a) cos θ tan θ = sin θ

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b) sin θ tan θ + cos θ = sec θ

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c) sin θ cos θ = sec θ csc θ

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4) For each of the identities you were able to verify in (3), show algebraically that the LHS = RHS by starting on one side and working to derive the other side.

a)

b)

2 ��� Class Packet #4.nb

Do it Yourself

5) Write each of the following in terms of sin θ and cos θ only.

a) sec θ cot θ

b) cos θ sec θ

c) cot θ - csc θ

6) Simplify each of the expressions given below (leave your answer in terms of sin θ or cos θ only).

a) sin θ cos θ

- cos θ sin θ

b) (3 sin θ - 2) (5 cos θ - 4)

Class Packet #4.nb ���3

7) For each of the following, use your calculator to determine whether or not the given equation appears to be true. Copy down the window that you used to view the graphs.

a) cos θ csc θ tan θ = 1

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b) sinθ cscθ

= sin2 θ

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c) tan θ - 1

sec θ = cot θ csc θ

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8) For each of the identities you were able to verify in (3), show algebraically that the LHS = RHS by starting on one side and working to derive the other side.

a)

b)

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