Pre-Calculus: Trigonometry functions

Anonymous
timer Asked: Dec 2nd, 2017
account_balance_wallet $20

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.

Tutor Answer

jesusale932
School: UT Austin

Here is my answer :)

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...

flag Report DMCA
Review

Anonymous
Outstanding Job!!!!

Similar Questions
Hot Questions
Related Tags

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors