Question 1

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

The secant function is the reciprocal of the cosine function. Therefore, $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{4}} = 4$. Since $\sin \theta > 0$, $\theta$ is in the first quadrant where secant is positive.

Question 2

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

The cosecant function is the reciprocal of the sine function, so $\csc \theta = \frac{1}{\sin \theta}$. Given $\sin \theta = - \frac{2}{3}$, $\csc \theta = - \frac{3}{2}$.

Question 3

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

In quadrant III, both sine and cosine are negative. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we can find $\cos \theta$. Given $\sin \theta = -\frac{5}{13}$, we have $(- \frac{5}{13})^2 + \cos^2 \theta = 1$. Solving for $\cos^2 \theta$ gives $\cos^2 \theta = 1 - (\frac{5}{13})^2 = \frac{169}{169} - \frac{25}{169} = \frac{144}{169}$. Taking the square root gives $\cos \theta = \pm \frac{12}{13}$, but since $\theta$ is in quadrant III where cosine is negative, $\cos \theta = -\frac{12}{13}$.

Question 4

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

In quadrant III, both sine and cosine are negative. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we can find $\sin \theta$. Given $\cos \theta = - \frac{\sqrt{11}}{6}$, we have $\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(- \frac{\sqrt{11}}{6}\right)^2 = 1 - \frac{11}{36} = \frac{25}{36}$. Since $\theta$ is in quadrant III, $\sin \theta = -\sqrt{\frac{25}{36}} = -\frac{5}{6}$.

Question 5

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

Given $\cot \theta = -6$, we can express this as $\frac{1}{\tan \theta} = -6$, implying $\tan \theta = -\frac{1}{6}$. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, and knowing $\tan \theta = \frac{\sin \theta}{\cos \theta}$, we can find $\sin \theta$ and $\cos \theta$. Since $\cos \theta < 0$ and we're in a quadrant where $\cos \theta$ is negative, we use the identity $\sin^2 \theta = 1 - \cos^2 \theta$ and the relationship $\tan^2 \theta + 1 = \frac{1}{\cos^2 \theta}$ to solve for $\sin \theta$. With $\tan \theta = -\frac{1}{6}$, we find $\sin \theta = \frac{\sqrt{37}}{37}$, considering the negative cosine value indicates we're in a quadrant where sine is positive.

Question 6

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

In quadrant IV, cosine is positive. Since $\csc \theta = -\frac{5}{3}$, $\sin \theta = -\frac{3}{5}$. Using $\sin^2 \theta + \cos^2 \theta = 1$, we find $\cos \theta = \frac{4}{5}$.

Question 7

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 8

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 9

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

In quadrant IV, $\cos \theta > 0$ and $\sin \theta < 0$, so $\tan \theta = \frac{\sin \theta}{\cos \theta}$ will be negative. Given $\cos \theta = \frac{1}{4}$, we can find $\sin \theta$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Substituting $\cos \theta = \frac{1}{4}$ gives $\sin^2 \theta = 1 - \left(\frac{1}{4}\right)^2 = 1 - \frac{1}{16} = \frac{15}{16}$. Taking the square root, $\sin \theta = \pm \sqrt{\frac{15}{16}}$, but since $\theta$ is in quadrant IV, $\sin \theta < 0$, so $\sin \theta = -\sqrt{\frac{15}{16}} = -\frac{\sqrt{15}}{4}$. Therefore, $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{15}}{4}}{\frac{1}{4}} = -\sqrt{15}$.

Question 10

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

In quadrant II, sine is positive. Using the identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$, and knowing $\cot \theta = -\sqrt{15}$, we have $\cos \theta = -\sqrt{15} \sin \theta$. Using $\sin^2 \theta + \cos^2 \theta = 1$, solve for $\sin \theta$ and find $\csc \theta = \frac{1}{\sin \theta} = 4$.

Question 11

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 12

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 13

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 14

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 15

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 16

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 17

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 18

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 19

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 20

Use the fundamental identities to find the value of the trigonometric function.&#10;

Accepted Answer

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Question 21

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

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Question 22

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

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Question 23

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.&#10; &#10;A) f(-x) = -f(x)&#10;B) f(-x) = f(x)

Accepted Answer

The answer of For the graph of a circular function...

Question 24

Perform the transformation.&#10;

Accepted Answer

The answer of Perform the transformation.&#10; ...

Question 25

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.&#10; &#10;A) f(-x) = -f(x)&#10;B) f(-x) = f(x)

Accepted Answer

The answer of For the graph of a circular function...

Question 26

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.&#10; &#10;A) f(-x) = -f(x)&#10;B) f(-x) = f(x)

Accepted Answer

The answer of For the graph of a circular function...

Question 27

Perform the transformation.&#10;

Accepted Answer

The answer of Perform the transformation.&#10; ...

Question 28

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.&#10; &#10;A) f(-x) = -f(x)&#10;B) f(-x) = f(x)

Accepted Answer

The answer of For the graph of a circular function...

Question 29

Perform the transformation.&#10;

Accepted Answer

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Question 30

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

The answer of Complete the sentence so the result is...

Question 31

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

The answer of Write the expression in terms of sine...

Question 32

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.&#10; &#10;A) f(-x) = -f(x)&#10;B) f(-x) = f(x)

Accepted Answer

The answer of For the graph of a circular function...

Question 33

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.&#10; &#10;A) f(-x) = -f(x)&#10;B) f(-x) = f(x)

Accepted Answer

The answer of For the graph of a circular function...

Question 34

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

The answer of Complete the sentence so the result is...

Question 35

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

The answer of Complete the sentence so the result is...

Question 36

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

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Question 37

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

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Question 38

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

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Question 39

Complete the sentence so the result is an identity. Let x be any real number.&#10;

Accepted Answer

The answer of Complete the sentence so the result is...

Question 40

Perform the transformation.&#10;

Accepted Answer

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Question 41

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Not an Identity&#10;B) Identity

Accepted Answer

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Question 42

Perform the indicated operations and simplify the result so there are no quotients.&#10;

Accepted Answer

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Question 43

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Not an identity&#10;B) Identity

Accepted Answer

The answer of Use a graphing calculator to make a...

Question 44

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 45

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 46

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Identity&#10;B) Not an identity

Accepted Answer

The answer of Use a graphing calculator to make a...

Question 47

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Identity&#10;B) Not an Identity

Accepted Answer

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Question 48

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 49

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Not an identity&#10;B) Identity

Accepted Answer

The answer of Use a graphing calculator to make a...

Question 50

Perform the indicated operations and simplify the result so there are no quotients.&#10;

Accepted Answer

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Question 51

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 52

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Not an identity&#10;B) Identity

Accepted Answer

The answer of Use a graphing calculator to make a...

Question 53

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10; &#10;A) Identity&#10;B) Not an Identity

Accepted Answer

The answer of Use a graphing calculator to make a...

Question 54

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 55

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 56

Perform the indicated operations and simplify the result so there are no quotients.&#10;

Accepted Answer

The answer of Perform the indicated operations and simplify the...

Question 57

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 58

Perform the indicated operations and simplify the result so there are no quotients.&#10;

Accepted Answer

The answer of Perform the indicated operations and simplify the...

Question 59

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 60

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.&#10;

Accepted Answer

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Question 61

Use the fundamental identities to simplify the expression.&#10;

Accepted Answer

The answer of Use the fundamental identities to simplify the...

Question 62

Use the fundamental identities to simplify the expression.&#10;

Accepted Answer

The answer of Use the fundamental identities to simplify the...

Question 63

Factor the trigonometric expression and simplify.&#10;

Accepted Answer

The answer of Factor the trigonometric expression and simplify.&#10; ...

Question 64

Perform the indicated operations and simplify the result so there are no quotients.&#10;

Accepted Answer

The answer of Perform the indicated operations and simplify the...

Deck 8: Trigonometric Identities and Equations