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: Which of the following is the correct even-odd identity for $$\tan(-\theta)$$?

$$\tan(-\theta) = -\tan \theta$$

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

$$\cos(-\theta) = \cos \theta$$

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

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

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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