# Pre-Calculus: Trigonometry functions

*label*Mathematics

*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.

## Tutor Answer

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...

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