Question 1

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 2

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

Accepted Answer

In quadrant II, $\sec \theta = \frac{1}{\cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$. Given $\sec \theta = -\frac{\sqrt{53}}{7}$, we find $\cos \theta = -\frac{7}{\sqrt{53}}$. To find $\sin \theta$, use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Substituting $\cos \theta$ gives $\sin^2 \theta = 1 - \left(-\frac{7}{\sqrt{53}}\right)^2$, leading to $\sin \theta = \sqrt{1 - \frac{49}{53}} = \frac{\sqrt{4}}{\sqrt{53}} = \frac{2}{\sqrt{53}}$. Since $\theta$ is in quadrant II, $\sin \theta > 0$, so $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{7}{\sqrt{53}}}{\frac{2}{\sqrt{53}}} = -\frac{7}{2}$.

Question 3

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

Accepted Answer

In quadrant III, both cosine and tangent are negative. Given $\tan \theta = \frac{2}{5}$, we can represent this as opposite/adjacent in a right triangle, implying sides of 2 (opposite) and -5 (adjacent) due to the negative value in quadrant III. Using the Pythagorean theorem, the hypotenuse $h = \sqrt{(-5)^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}$. Thus, $\cos \theta = \frac{adjacent}{hypotenuse} = \frac{-5}{\sqrt{29}}$, which simplifies to $- \frac { 5 \sqrt { 29 } } { 29 }$ when rationalized.

Question 4

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

Accepted Answer

In quadrant II, sine is positive and tangent is negative. Using the Pythagorean identity, $\cos^2 \theta = 1 - \sin^2 \theta$, we find $\cos \theta = -\sqrt{1 - \left(\frac{3}{4}\right)^2} = -\frac{\sqrt{7}}{4}$. Thus, $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{4}}{-\frac{\sqrt{7}}{4}} = -\frac{3\sqrt{7}}{7}$.

Question 5

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, if $\cos \theta = \frac{1}{4}$, then $\sec \theta = \frac{1}{\cos \theta} = 4$.

Question 6

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

Accepted Answer

In quadrant III, both sine and cosine are negative, making tangent positive. $\cot \theta = \frac{1}{\tan \theta} = \frac{3}{\sqrt{7}}$, which simplifies to $\frac{3\sqrt{7}}{7}$.

Question 7

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

Accepted Answer

Given $\sec \theta = \frac{\sqrt{5}}{2}$, we know $\cos \theta = \frac{2}{\sqrt{5}}$. Since $\sin \theta < 0$ and $\sec \theta > 0$, $\theta$ is in the fourth quadrant where $\cos \theta > 0$ and $\sin \theta < 0$. Using Pythagoras, $\sin^2 \theta + \cos^2 \theta = 1$, we find $\sin \theta = -\frac{1}{\sqrt{5}}$. Thus, $\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{1}{2}$.

Question 8

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 9

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

Accepted Answer

Given $\tan \theta = 5$, we can represent this as $\frac{\sin \theta}{\cos \theta} = 5$. Since $\sin \theta < 0$, we are in either the third or fourth quadrant where cosine is positive in the fourth quadrant and negative in the third. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, and knowing $\tan^2 \theta + 1 = \frac{1}{\cos^2 \theta}$, we find $\cos \theta = - \frac{\sqrt{26}}{26}$ because $\cos^2 \theta = \frac{1}{1 + \tan^2 \theta} = \frac{1}{1 + 25} = \frac{1}{26}$, and we choose the negative root due to the quadrant information.

Question 10

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

Accepted Answer

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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

Perform the transformation.&#10;

Accepted Answer

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

Question 22

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

Accepted Answer

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

Question 23

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 24

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

Accepted Answer

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

Question 25

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

Accepted Answer

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Question 26

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

Accepted Answer

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Question 27

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

Accepted Answer

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Question 28

Perform the transformation.&#10;

Accepted Answer

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Question 29

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 30

Perform the transformation.&#10;

Accepted Answer

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Question 31

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 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;

Accepted Answer

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

Question 33

Perform the transformation.&#10;

Accepted Answer

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Question 34

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

Accepted Answer

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Question 35

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

Accepted Answer

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Question 36

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 37

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

Accepted Answer

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

Question 38

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

Accepted Answer

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

Question 39

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

Accepted Answer

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

Question 40

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

Accepted Answer

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Question 42

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 43

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10;

Accepted Answer

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Question 44

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

Accepted Answer

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Question 47

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 48

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#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;

Accepted Answer

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Question 50

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 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

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

Accepted Answer

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Question 53

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 54

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

Accepted Answer

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Question 55

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10;

Accepted Answer

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Question 56

Use a graphing calculator to make a conjecture as to whether each equation is an identity.&#10;

Accepted Answer

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

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

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 59

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

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Question 62

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

Accepted Answer

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Question 63

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

Accepted Answer

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Question 64

Factor the trigonometric expression and simplify.&#10;

Accepted Answer

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

Deck 7: Trigonometric Identities and Equations