Question 1

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

Accepted Answer

$\cot \theta \sec \theta \sin \theta = \frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \sin \theta = 1$, since $\frac{\cos \theta}{\sin \theta} \cdot \sin \theta = \cos \theta$ and $\frac{1}{\cos \theta} \cdot \cos \theta = 1$.

Question 2

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Accepted Answer

Given $\cos \theta = \frac{1}{6}$ and $\sin \theta < 0$, we can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\sin \theta$. Substituting $\cos \theta = \frac{1}{6}$, we get $\sin^2 \theta + \left(\frac{1}{6}\right)^2 = 1$, which simplifies to $\sin^2 \theta = 1 - \frac{1}{36} = \frac{35}{36}$. Since $\sin \theta < 0$, $\sin \theta = -\sqrt{\frac{35}{36}} = -\frac{\sqrt{35}}{6}$. Therefore, $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{35}}{6}}{\frac{1}{6}} = -\sqrt{35}$.

Question 3

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

Accepted Answer

The expression $\frac { \tan \theta } { \cot \theta }$ simplifies to $\tan^2 \theta$ because $\cot \theta$ is the reciprocal of $\tan \theta$, so dividing $\tan \theta$ by $\cot \theta$ is equivalent to $\tan \theta \times \tan \theta = \tan^2 \theta$.

Question 4

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

$\sin(-x) \csc x = \frac{-\sin x}{\sin x} = -1$, because $\sin(-x) = -\sin x$ and $\csc x = 1/\sin x$.

Question 5

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

Accepted Answer

The expression $\frac { \tan \theta } { \sec \theta }$ simplifies to $\sin \theta$ because $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$, so dividing $\tan \theta$ by $\sec \theta$ gives $\sin \theta$.

Question 6

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Accepted Answer

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Question 7

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

Accepted Answer

The expression can be simplified using the identity $\tan^2 \theta + 1 = \sec^2 \theta$ and the fact that $\csc^2 \theta = 1 + \cot^2 \theta$. Since $\tan^2 \theta = \sec^2 \theta - 1$, when you multiply $\tan^2 \theta$ by $\csc^2 \theta$, you essentially get $\sec^2 \theta$ because $\csc^2 \theta$ can be rewritten as $1/\sin^2 \theta$, and $\tan^2 \theta$ involves $\sin^2 \theta$ in its denominator when expressed in terms of sine and cosine. Therefore, the correct simplification is $\sec^2 \theta$.

Question 8

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

Accepted Answer

The expression simplifies to $\csc ^ { 2 } \theta$ because $\frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta } = \cot ^ { 2 } \theta$ and $\csc \theta \sin \theta = 1$, so the expression becomes $\cot ^ { 2 } \theta + 1 = \csc ^ { 2 } \theta$ using the Pythagorean identity.

Question 9

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Accepted Answer

Given $\cot \theta = -\sqrt{35}$, we can use the identity $\cot^2 \theta + 1 = \csc^2 \theta$ to find $\csc \theta$. Substituting $\cot \theta = -\sqrt{35}$ gives $(- \sqrt{35})^2 + 1 = \csc^2 \theta$, which simplifies to $35 + 1 = 36 = \csc^2 \theta$. Taking the square root gives $\csc \theta = \pm 6$. Since $\cos \theta < 0$ and $\cot \theta < 0$, we are in the second or fourth quadrant. However, $\csc \theta$ is positive in the second quadrant and negative in the fourth. Given that $\cot \theta$ and $\cos \theta$ indicate we're in a quadrant where $\csc \theta$ can only be positive (since $\cot \theta$ being negative suggests we're not in the first or third quadrant where $\cos \theta$ and $\sin \theta$ have the same signs), the correct answer is $\csc \theta = 6$.

Question 10

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

Accepted Answer

The expression simplifies to $\sec ^ { 2 } \theta$ because $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1$ (a basic trigonometric identity), and $\tan ^ { 2 } \theta + 1 = \sec ^ { 2 } \theta$. So, the expression becomes $1 + \tan ^ { 2 } \theta = \sec ^ { 2 } \theta$.

Question 11

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Accepted Answer

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Question 12

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

Accepted Answer

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Question 13

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Accepted Answer

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Question 14

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Accepted Answer

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Question 15

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Accepted Answer

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Question 16

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

Accepted Answer

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Question 17

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

Accepted Answer

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Question 18

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

Accepted Answer

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Question 19

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Accepted Answer

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Question 20

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Accepted Answer

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Question 21

Simplify the expression.&#10;

Accepted Answer

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Question 22

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

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Question 23

Simplify the expression.&#10;

Accepted Answer

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Question 24

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

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Question 25

Simplify the expression.&#10;

Accepted Answer

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Question 26

Simplify the expression to either 1 or -1.&#10;sec (-x) cos (-x)&#10;A) -1&#10;B) 1

Accepted Answer

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Question 27

Simplify the expression.&#10;

Accepted Answer

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Question 28

Simplify the expression to either 1 or -1.&#10;csc (-x) sin (-x)&#10;A) -1&#10;B) 1

Accepted Answer

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Question 29

Simplify the expression.&#10;

Accepted Answer

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Question 30

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

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Question 31

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

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Question 32

Simplify the expression.&#10;

Accepted Answer

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Question 33

Simplify the expression.&#10;

Accepted Answer

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Question 34

Simplify the expression.&#10;

Accepted Answer

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Question 35

Simplify the expression.&#10;

Accepted Answer

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Question 36

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

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Question 37

Simplify the expression.&#10;

Accepted Answer

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Question 38

Simplify the expression to either 1 or -1.&#10;

Accepted Answer

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Question 39

Simplify the expression.&#10;

Accepted Answer

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Question 40

Simplify the expression.&#10;

Accepted Answer

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Question 41

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 42

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 43

Simplify the expression.&#10;

Accepted Answer

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Question 44

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 45

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 46

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 47

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 48

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 49

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 50

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 51

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 52

Simplify the expression.&#10;

Accepted Answer

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Question 53

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 54

Simplify the expression.&#10;

Accepted Answer

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Question 55

Simplify the expression.&#10;

Accepted Answer

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Question 56

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 57

Simplify the expression.&#10;

Accepted Answer

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Question 58

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 59

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 60

Write each expression in factored form as an algebraic expression of a single trigonometric function.&#10;

Accepted Answer

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Question 61

Find all solutions in the interval [0, 2&#10;

Accepted Answer

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Question 62

Determine if the following is an identity.&#10;

Accepted Answer

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Question 63

Determine if the following is an identity.&#10;

Accepted Answer

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Question 64

Find all solutions to the equation.&#10;

Accepted Answer

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Deck 5: Analytic Trigonometry