Q: Which of the following is the correct reciprocal identity for $\csc \theta$?

$\csc \theta = \dfrac{1}{\sin \theta}$

Q: Which of the following is the correct Pythagorean identity?

$\sin^2 \theta + \cos^2 \theta = 1$

Q: Express $\tan \theta + \cot \theta$ in terms of $\sin \theta$ and $\cos \theta$, and simplify so that no quotients appear.

$\dfrac{\sin \theta}{\cos \theta} + \dfrac{\cos \theta}{\sin \theta} = \dfrac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \dfrac{1}{\sin \theta \cos \theta}$

Q: Write $\cos x$ in terms of $\tan x$ using a Pythagorean identity.

$\cos x = \dfrac{1}{\sqrt{1 + \tan^2 x}}$

Q: Express $\tan \theta$ in terms of $\cos \theta$ only.

$\tan \theta = \dfrac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}$

Q: Which of the following is the correct quotient identity for $\cot \theta$?

$\cot \theta = \dfrac{\cos \theta}{\sin \theta}$

Q: Which of the following is the correct reciprocal identity for $\sec \theta$?

$\sec \theta = \dfrac{1}{\cos \theta}$

Q: If $\theta$ is a real number, which of the following is always true?

$\sin^2 \theta + \cos^2 \theta = 1$

Q: Which of the following is the correct Pythagorean identity involving $\cot \theta$ and $\csc \theta$?

$1 + \cot^2 \theta = \csc^2 \theta$

Question 1

Which of the following is the correct reciprocal identity for $\csc 	heta$?

Accepted Answer

$\csc 	heta = \dfrac{1}{\sin 	heta}$

Question 2

Given $\sin 	heta = \dfrac{5}{8}$ and $	heta$ is in quadrant IV, what is $\cos 	heta$?

Accepted Answer

$-\dfrac{3}{8}$

Question 3

Which of the following is the correct Pythagorean identity?

Accepted Answer

$\sin^2 	heta + \cos^2 	heta = 1$

Question 4

Express $	an 	heta + \cot 	heta$ in terms of $\sin 	heta$ and $\cos 	heta$, and simplify so that no quotients appear.

Accepted Answer

$\dfrac{\sin 	heta}{\cos 	heta} + \dfrac{\cos 	heta}{\sin 	heta} = \dfrac{\sin^2 	heta + \cos^2 	heta}{\sin 	heta \cos 	heta} = \dfrac{1}{\sin 	heta \cos 	heta}$

Question 5

If $	an 	heta = -2$ and $	heta$ is in quadrant IV, what is $\sec 	heta$?

Accepted Answer

$-\dfrac{\sqrt{5}}{1}$

Question 6

Which of the following is the correct even-odd identity for $	an(-	heta)$?

Accepted Answer

$	an(-	heta) = -	an 	heta$

Question 7

Write $\cos x$ in terms of $	an x$ using a Pythagorean identity.

Accepted Answer

$\cos x = \dfrac{1}{\sqrt{1 + 	an^2 x}}$

Question 8

Express $	an 	heta$ in terms of $\cos 	heta$ only.

Accepted Answer

$	an 	heta = \dfrac{\sqrt{1 - \cos^2 	heta}}{\cos 	heta}$

Question 9

Given $\sin 	heta = \dfrac{5}{8}$ and $	heta$ is in quadrant IV, what is $	an 	heta$?

Accepted Answer

$-\dfrac{5}{3}$

Question 10

Which of the following is the correct quotient identity for $\cot 	heta$?

Accepted Answer

$\cot 	heta = \dfrac{\cos 	heta}{\sin 	heta}$

Question 11

Simplify $\dfrac{1 + 	an^2 	heta}{1 - \sec^2 	heta}$ in terms of $\sin 	heta$ and $\cos 	heta$.

Accepted Answer

$-1$

Question 12

If $	heta$ is an angle in quadrant IV, which trigonometric functions are negative?

Accepted Answer

Sine and tangent

Question 13

Which of the following is the correct reciprocal identity for $\sec 	heta$?

Accepted Answer

$\sec 	heta = \dfrac{1}{\cos 	heta}$

Question 14

Which of the following is the correct even-odd identity for $\cos(-	heta)$?

Accepted Answer

$\cos(-	heta) = \cos 	heta$

Question 15

A medical imaging device rotates around a patient, measuring angles $	heta$ in degrees. If $\sin 	heta = 0.6$ and $	heta$ is in quadrant II, what is $\cos 	heta$?

Accepted Answer

$-0.8$

Question 16

If $	heta$ is a real number, which of the following is always true?

Accepted Answer

$\sin^2 	heta + \cos^2 	heta = 1$

Question 17

Express $\sec(-	heta)$ in terms of $\sec 	heta$.

Accepted Answer

$\sec(-	heta) = \sec 	heta$

Question 18

A satellite orbits Earth and its position is modeled by $	heta$ in radians. If $	an 	heta = 3$ and $	heta$ is in quadrant III, what is $\sin 	heta$?

Accepted Answer

$-\dfrac{3}{\sqrt{10}}$

Question 19

Which of the following is the correct Pythagorean identity involving $\cot 	heta$ and $\csc 	heta$?

Accepted Answer

$1 + \cot^2 	heta = \csc^2 	heta$

Fundamental Trigonometric Identities and Their Applications

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