Basic Calculus Module 4: Trigonometry

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Questions and Answers

What is the Pythagorean Identity expressed in terms of sine and cosine?

$ ext{sin}^2 heta - ext{tan}^2 heta = 1$
$ ext{sin}^2 heta + ext{cos}^2 heta = 1$ (correct)
$ ext{sin}^2 heta - ext{cos}^2 heta = 0$
$ ext{sin}^2 heta + ext{cos}^2 heta = ext{tan}^2 heta$

In which quadrant are both sine and cosine positive?

Quadrant I (correct)
Quadrant II
Quadrant IV
Quadrant III

What is the value of sin(60°)?

$rac{√3}{2}$ (correct)
$rac{√2}{2}$
$rac{1}{2}$
$1$

How is the Pythagorean Identity used in trigonometry?

To simplify expressions containing sine and cosine. (D) Signup and view all the answers

Which of the following represents a common Pythagorean identity?

$sec^2(\theta) - tan^2(\theta) = 1$ (C) Signup and view all the answers

Given that $sin(\theta) \neq 0$, which identity is valid?

$cot^2(\theta) = csc^2(\theta) - 1$ (B), $1 + cot^2(\theta) = csc^2(\theta)$ (C) Signup and view all the answers

When verifying the identity $(sec(\theta) - tan(\theta))(sec(\theta) + tan(\theta)) = 1$, which step involves an identity?

Expand to $sec^2(\theta) - tan^2(\theta)$ (D) Signup and view all the answers

What transformation verifies the identity $tan(\theta) = sin(\theta) sec(\theta)$?

Setting $sec(\theta) = \frac{1}{cos(\theta)}$ (A), Multiplying both sides by $cos(\theta)$ (B) Signup and view all the answers

Which of the following identities is not a standard Pythagorean identity?

$tan^2(\theta) + sec^2(\theta) = 1$ (D) Signup and view all the answers

If $sec(\theta) = \frac{1}{cos(\theta)}$, how can you express $tan(\theta)$?

$tan(\theta) = sin(\theta) sec(\theta)$ (A) Signup and view all the answers

How would you express the relationship $5sec(\theta)tan(\theta)$ using Pythagorean identities?

$5sec^2(\theta)$ (A) Signup and view all the answers

What is the correct identity involving $csc(\theta)$ and $cot(\theta)$?

$csc^2(\theta) = 1 + cot^2(\theta)$ (D) Signup and view all the answers

Which of the following represents a common alternate form of the Pythagorean identity?

𝑐𝑜𝑠2 (𝜃) = 1 − 𝑠𝑖𝑛2 (𝜃) (A) Signup and view all the answers

What is the identity derived from dividing the Pythagorean identity by cosine squared?

1 + 𝑡𝑎𝑛2 (𝜃) = 𝑠𝑒𝑐2 (𝜃) (C) Signup and view all the answers

Which Pythagorean identity requires that sin(𝜃) ≠ 0 for its derivation?

𝑐𝑜𝑡2 (𝜃) + 1 = 𝑐𝑠𝑐2 (𝜃) (B) Signup and view all the answers

Which of the following is NOT a Pythagorean identity?

𝑐𝑜𝑠2 (𝜃) + 𝑐𝑜𝑡2 (𝜃) = 1 (C) Signup and view all the answers

What is the correct form for the Quadrant II solution given in the content?

𝜃 = rac{3𝜋}{4} + 2𝜋𝑘 (B) Signup and view all the answers

Which of the following identities can be derived from the Pythagorean identity by dividing by sine squared?

1 + 𝑐𝑜𝑡2 (𝜃) = 𝑐𝑠𝑐2 (𝜃) (C) Signup and view all the answers

What can be combined from the Quadrant II and Quadrant IV solutions?

𝜃 = rac{3𝜋}{4} + 𝜋𝑘 (C) Signup and view all the answers

If $𝑐𝑜𝑠(𝜃) ≠ 0$, what is the result of dividing the Pythagorean identity by $𝑐𝑜𝑠2 (𝜃)$?

1 + 𝑡𝑎𝑛2 (𝜃) = 𝑠𝑒𝑐2 (𝜃) (C) Signup and view all the answers

Flashcards

Pythagorean Identity

The expression that relates the trigonometric functions sine, cosine, and tangent. It states that the square of the cosine of an angle plus the square of the sine of the angle equals 1. Specifically, for any angle, 𝑐𝑜𝑠 2 (𝜃) + 𝑠𝑖𝑛2 (𝜃) = 1.

Cosine (cos)

A trigonometric function that represents the ratio of the adjacent side of a right triangle to the hypotenuse. It is also defined as the reciprocal of the sine function.

Sine (sin)

A trigonometric function representing the ratio of the opposite side to the hypotenuse in a right triangle. It's also defined as the reciprocal of the cosecant function.

Tangent (tan)

A trigonometric function that defines the ratio of the opposite side to the adjacent side of a right triangle. It is also defined as the quotient of sine and cosine.