Solved on Jan 22, 2024

Find the cosecant of $\frac{5 \pi}{3}$ .

STEP 1

Assumptions
1. We need to find the value of $\csc \frac{5 \pi}{3}$ .
2. The cosecant function ( $\csc$ ) is the reciprocal of the sine function ( $\sin$ ).
3. We will use the unit circle and the properties of trigonometric functions to find the value.

STEP 2

Recall the definition of the cosecant function in terms of the sine function.
$\csc \theta = \frac{1}{\sin \theta}$

STEP 3

To find the value of $\csc \frac{5 \pi}{3}$ , we first need to find the value of $\sin \frac{5 \pi}{3}$ .

STEP 4

The angle $\frac{5 \pi}{3}$ is an angle in standard position whose terminal side is in the fourth quadrant of the unit circle.

STEP 5

Since $\frac{5 \pi}{3}$ is more than $2\pi$ , we can subtract $2\pi$ to find a coterminal angle between $0$ and $2\pi$ .
$\frac{5 \pi}{3} - 2\pi = \frac{5 \pi}{3} - \frac{6 \pi}{3} = -\frac{\pi}{3}$

STEP 6

The sine function is an odd function, which means $\sin(-\theta) = -\sin(\theta)$ . Therefore, $\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3}$ .

STEP 7

Find the sine of the reference angle $\frac{\pi}{3}$ .

STEP 8

The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$ .

STEP 9

Apply the result from STEP_6 to find $\sin \frac{5 \pi}{3}$ .
$\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}$

STEP 10

Now that we have the value of $\sin \frac{5 \pi}{3}$ , we can find the cosecant of the angle by taking the reciprocal.
$\csc \frac{5 \pi}{3} = \frac{1}{\sin \frac{5 \pi}{3}}$

STEP 11

Substitute the value of $\sin \frac{5 \pi}{3}$ into the expression.
$\csc \frac{5 \pi}{3} = \frac{1}{-\frac{\sqrt{3}}{2}}$

STEP 12

Calculate the value of $\csc \frac{5 \pi}{3}$ .
$\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}}$

STEP 13

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$ .
$\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

STEP 14

Simplify the expression.
$\csc \frac{5 \pi}{3} = -\frac{2\sqrt{3}}{3}$
The value of $\csc \frac{5 \pi}{3}$ is $-\frac{2\sqrt{3}}{3}$ .

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Find the cosecant of $\frac{5 \pi}{3}$ .

STEP 1

Assumptions
1. We need to find the value of $\csc \frac{5 \pi}{3}$ .
2. The cosecant function ( $\csc$ ) is the reciprocal of the sine function ( $\sin$ ).
3. We will use the unit circle and the properties of trigonometric functions to find the value.

STEP 2

Recall the definition of the cosecant function in terms of the sine function.
$\csc \theta = \frac{1}{\sin \theta}$

STEP 3

To find the value of $\csc \frac{5 \pi}{3}$ , we first need to find the value of $\sin \frac{5 \pi}{3}$ .

STEP 4

The angle $\frac{5 \pi}{3}$ is an angle in standard position whose terminal side is in the fourth quadrant of the unit circle.

STEP 5

Since $\frac{5 \pi}{3}$ is more than $2\pi$ , we can subtract $2\pi$ to find a coterminal angle between $0$ and $2\pi$ .
$\frac{5 \pi}{3} - 2\pi = \frac{5 \pi}{3} - \frac{6 \pi}{3} = -\frac{\pi}{3}$

STEP 6

The sine function is an odd function, which means $\sin(-\theta) = -\sin(\theta)$ . Therefore, $\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3}$ .

STEP 7

Find the sine of the reference angle $\frac{\pi}{3}$ .

STEP 8

The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$ .

STEP 9

Apply the result from STEP_6 to find $\sin \frac{5 \pi}{3}$ .
$\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}$

STEP 10

Now that we have the value of $\sin \frac{5 \pi}{3}$ , we can find the cosecant of the angle by taking the reciprocal.
$\csc \frac{5 \pi}{3} = \frac{1}{\sin \frac{5 \pi}{3}}$

STEP 11

Substitute the value of $\sin \frac{5 \pi}{3}$ into the expression.
$\csc \frac{5 \pi}{3} = \frac{1}{-\frac{\sqrt{3}}{2}}$

STEP 12

Calculate the value of $\csc \frac{5 \pi}{3}$ .
$\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}}$

STEP 13

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$ .
$\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

STEP 14

Simplify the expression.
$\csc \frac{5 \pi}{3} = -\frac{2\sqrt{3}}{3}$
The value of $\csc \frac{5 \pi}{3}$ is $-\frac{2\sqrt{3}}{3}$ .

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Find the cosecant of 5π3\frac{5 \pi}{3}35π​.

STEP 1

Assumptions1. We need to find the value of csc⁡5π3\csc \frac{5 \pi}{3}csc35π​.2. The cosecant function (csc⁡\csccsc) is the reciprocal of the sine function (sin⁡\sinsin).3. We will use the unit circle and the properties of trigonometric functions to find the value.

STEP 2

Recall the definition of the cosecant function in terms of the sine function.csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}cscθ=sinθ1​

STEP 3

To find the value of csc⁡5π3\csc \frac{5 \pi}{3}csc35π​, we first need to find the value of sin⁡5π3\sin \frac{5 \pi}{3}sin35π​.

STEP 4

The angle 5π3\frac{5 \pi}{3}35π​ is an angle in standard position whose terminal side is in the fourth quadrant of the unit circle.

STEP 5

Since 5π3\frac{5 \pi}{3}35π​ is more than 2π2\pi2π, we can subtract 2π2\pi2π to find a coterminal angle between 000 and 2π2\pi2π.5π3−2π=5π3−6π3=−π3\frac{5 \pi}{3} - 2\pi = \frac{5 \pi}{3} - \frac{6 \pi}{3} = -\frac{\pi}{3}35π​−2π=35π​−36π​=−3π​

STEP 6

The sine function is an odd function, which means sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)sin(−θ)=−sin(θ). Therefore, sin⁡5π3=−sin⁡π3\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3}sin35π​=−sin3π​.

STEP 7

Find the sine of the reference angle π3\frac{\pi}{3}3π​.

STEP 8

The sine of π3\frac{\pi}{3}3π​ is 32\frac{\sqrt{3}}{2}23​​.

STEP 9

Apply the result from STEP_6 to find sin⁡5π3\sin \frac{5 \pi}{3}sin35π​.sin⁡5π3=−sin⁡π3=−32\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}sin35π​=−sin3π​=−23​​

STEP 10

Now that we have the value of sin⁡5π3\sin \frac{5 \pi}{3}sin35π​, we can find the cosecant of the angle by taking the reciprocal.csc⁡5π3=1sin⁡5π3\csc \frac{5 \pi}{3} = \frac{1}{\sin \frac{5 \pi}{3}}csc35π​=sin35π​1​

STEP 11

Substitute the value of sin⁡5π3\sin \frac{5 \pi}{3}sin35π​ into the expression.csc⁡5π3=1−32\csc \frac{5 \pi}{3} = \frac{1}{-\frac{\sqrt{3}}{2}}csc35π​=−23​​1​

STEP 12

Calculate the value of csc⁡5π3\csc \frac{5 \pi}{3}csc35π​.csc⁡5π3=−23\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}}csc35π​=−3​2​

STEP 13

To rationalize the denominator, multiply the numerator and denominator by 3\sqrt{3}3​.csc⁡5π3=−23⋅33\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}csc35π​=−3​2​⋅3​3​​

STEP 14

Simplify the expression.csc⁡5π3=−233\csc \frac{5 \pi}{3} = -\frac{2\sqrt{3}}{3}csc35π​=−323​​The value of csc⁡5π3\csc \frac{5 \pi}{3}csc35π​ is −233-\frac{2\sqrt{3}}{3}−323​​.

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Find the cosecant of $\frac{5 \pi}{3}$ .

Assumptions
1. We need to find the value of $\csc \frac{5 \pi}{3}$ .
2. The cosecant function ( $\csc$ ) is the reciprocal of the sine function ( $\sin$ ).
3. We will use the unit circle and the properties of trigonometric functions to find the value.

Recall the definition of the cosecant function in terms of the sine function.
$\csc \theta = \frac{1}{\sin \theta}$

To find the value of $\csc \frac{5 \pi}{3}$ , we first need to find the value of $\sin \frac{5 \pi}{3}$ .

The angle $\frac{5 \pi}{3}$ is an angle in standard position whose terminal side is in the fourth quadrant of the unit circle.

Since $\frac{5 \pi}{3}$ is more than $2\pi$ , we can subtract $2\pi$ to find a coterminal angle between $0$ and $2\pi$ .
$\frac{5 \pi}{3} - 2\pi = \frac{5 \pi}{3} - \frac{6 \pi}{3} = -\frac{\pi}{3}$

The sine function is an odd function, which means $\sin(-\theta) = -\sin(\theta)$ . Therefore, $\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3}$ .

Find the sine of the reference angle $\frac{\pi}{3}$ .

The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$ .

Apply the result from STEP_6 to find $\sin \frac{5 \pi}{3}$ .
$\sin \frac{5 \pi}{3} = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}$

Now that we have the value of $\sin \frac{5 \pi}{3}$ , we can find the cosecant of the angle by taking the reciprocal.
$\csc \frac{5 \pi}{3} = \frac{1}{\sin \frac{5 \pi}{3}}$

Substitute the value of $\sin \frac{5 \pi}{3}$ into the expression.
$\csc \frac{5 \pi}{3} = \frac{1}{-\frac{\sqrt{3}}{2}}$

Calculate the value of $\csc \frac{5 \pi}{3}$ .
$\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}}$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$ .
$\csc \frac{5 \pi}{3} = -\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Simplify the expression.
$\csc \frac{5 \pi}{3} = -\frac{2\sqrt{3}}{3}$
The value of $\csc \frac{5 \pi}{3}$ is $-\frac{2\sqrt{3}}{3}$ .