# Pre-Calculus: Trigonometry functions

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Question description

Five questions involving: Fundamental Identities, Trigonometric Functions, Trigonometric Equations, Sum and Difference Formulas, and Multiple Angle Formulas.

Question 1: Using Fundamental Identities Use the Pythagorean identity sin2 Θ + cos2 Θ = 1 to derive the other Pythagorean identities, 1 + tan2 Θ = sec2 Θ and 1 + cot2 Θ = csc2 Θ. Discuss how to remember these identities and other fundamental identities. Question 2: Verify Trigonometric Functions For each question: A. Verify the identity. B. Determine if the identity is true for the given value of x. Explain. Question 3: Trigonometric Equations 1. Describe the difference between verifying a trigonometric identity and solving a trigonometric equation. 2. Describe several techniques for solving trigonometric equations. Question 4: Sum and Difference Formulas 1. You can write the sum and difference formulas for cosine as a single equation: cos (u ± v) = cos u cos v ∓ sin u sin v. Explain why the symbol ± is used on the left side, but the symbol ∓ is used on the right side. Then use the symbols ± and ∓ to write the sum and difference formulas for sine and tangent as single equations. Question 5: Multiple Angle Formulas 1. Describe how you can use a double-angle formula or a half-angle formula to derive the formula for the area of an isosceles triangle. Use a labeled sketch to illustrate your derivation. Then write two examples that show how your formula can be used.

jesusale932
School: UT Austin

Question 1: Using Fundamental Identities
Use the Pythagorean identity sin2 Θ + cos2 Θ = 1 to derive the other Pythagorean identities, 1 + tan2 Θ =
sec2 Θ and 1 + cot2 Θ = csc2 Θ. Discuss how to remember these identities and other fundamental
identities.

Pythagorean identity: 1 + tan2 Θ = sec2 Θ

If sin2 Θ + cos2 Θ = 1

sin2 θ + cos 2 θ = 1
Dividing by

1
cos2 𝜃

:
sin2 θ cos 2 θ
1
+
=
cos 2 𝜃 cos 2 𝜃 cos 2 𝜃

We know that

𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃

= 𝑡𝑎𝑛θ and

1
𝑐𝑜𝑠𝜃

= 𝑠𝑒𝑐θ so, substituting we get
𝐭𝐚𝐧𝟐 𝜽 + 𝟏 = 𝐬𝐞𝐜 𝟐 𝜽

Pythagorean identity: 1 + cot2 Θ = csc2 Θ

If sin2 Θ + cos2 Θ = 1

sin2 θ + cos 2 θ = 1
Dividing by

1
sin2 𝜃

:
sin2 θ cos 2 θ
1
+
=
2
2
sin 𝜃 sin 𝜃 sin2 𝜃

We know that

𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃

= 𝑐𝑜𝑡θ and

1
𝑠𝑖𝑛𝜃

= 𝑐𝑠𝑐θ so, substituting we get
𝟏 + 𝐜𝐨𝐭 𝟐 𝜽 = 𝐜𝐬𝐜 𝟐 𝜽

The most important is to know the fundamental trigonometric identities, sinθ, cosθ, and tanθ, and then we
can prove the other identities. We remember the Pythagorean identity sin2 Θ + cos2 Θ = 1 because it from
of a circumference of radius 1 and adjacent side equal to cos Θ and opposite side equal to sin Θ

Question 2: Verify Trigonometric Functions
For each question:
A. Verify the identity.
B. Determine if the identity is true for the given value of x. Explain.

sec 𝑥
tan 𝑥
=
tan 𝑥 sec 𝑥 − cos 𝑥
We know that
tan 𝑥 =

sin 𝑥
cos 𝑥

sec 𝑥 =

1
cos 𝑥

And,

Now, substituting in the expression
sec 𝑥
=
tan 𝑥

sin 𝑥
cos 𝑥
1
− cos 𝑥
cos 𝑥

sin 𝑥
sec 𝑥
cos
𝑥
=
tan 𝑥 1 − cos 2 𝑥
cos 𝑥
sec 𝑥
sin 𝑥 cos 𝑥
=
tan 𝑥 cos 𝑥 (1 − cos2 𝑥)
sec 𝑥
sin 𝑥
=
tan 𝑥 1 − cos 2 𝑥
But, we know Pythagorean identity is sin2 Θ + cos2 Θ = 1
So, 1 − cos 2 𝑥 = sin2 𝑥

Substituting
sec 𝑥
sin 𝑥
=
tan 𝑥 sin2 𝑥
sec 𝑥
1
=
tan 𝑥 sin 𝑥
sec 𝑥

Transforming tan 𝑥

Will be
1
cos 𝑥 = 1
𝑠𝑖𝑛 𝑥 sin 𝑥
cos 𝑥
cos 𝑥
1
=
cos 𝑥 sin 𝑥 sin 𝑥
1
1
=
sin 𝑥 sin 𝑥
sec 𝜋
tan 𝜋
=
tan 𝜋 sec 𝜋 − cos 𝜋

The identity is false for the given value of 𝜋 because the graph of sin is zero when x is 𝜋, so the result is
1
0

1

. Then 0 does not belong to the domain of the functionsin 𝑥.

cot 𝑥 − 1 1 − tan 𝑥
=
cot 𝑥 + 1 1 + tan 𝑥
We, know that
cot 𝑥 =

1
tan 𝑥

Substituting
1
tan 𝑥 − 1 = 1 − tan 𝑥
1
1 + tan 𝑥
tan 𝑥 + 1
1 − tan 𝑥
tan 𝑥 = 1 − tan 𝑥
1 + tan 𝑥 1 + tan 𝑥
tan 𝑥
tan 𝑥 (1 − tan 𝑥) 1 − tan 𝑥
=
tan 𝑥 (1 − tan 𝑥) 1 + tan 𝑥
1 − tan 𝑥 1 − tan 𝑥
=
1 + tan 𝑥 1 + tan 𝑥

𝜋
𝜋
cot 4 − 1 1...

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Anonymous
Outstanding Job!!!!

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