Math / Trigonometry

QuestionQuestion 7 - 5 marks What is the exact value of $\csc 300^{\circ}$ ? a) $\frac{\sqrt{3}}{2}$ b) $\frac{2}{\sqrt{3}}$ c) $-\frac{2 \sqrt{3}}{3}$ d) $\frac{1}{2}$

Studdy Solution

STEP 1

1. The angle $300^\circ$ is measured in standard position.
2. The cosecant function, $\csc \theta$ , is the reciprocal of the sine function, $\sin \theta$ .

STEP 2

1. Determine the reference angle for $300^\circ$ .
2. Identify the quadrant in which $300^\circ$ lies.
3. Calculate $\sin 300^\circ$ using the reference angle and quadrant information.
4. Find $\csc 300^\circ$ by taking the reciprocal of $\sin 300^\circ$ .
5. Choose the correct answer from the given options.

STEP 3

Determine the reference angle for $300^\circ$ .
The reference angle is calculated as:
$360^\circ - 300^\circ = 60^\circ$

STEP 4

Identify the quadrant in which $300^\circ$ lies.
Since $300^\circ$ is between $270^\circ$ and $360^\circ$ , it lies in the fourth quadrant.

STEP 5

Calculate $\sin 300^\circ$ using the reference angle and quadrant information.
In the fourth quadrant, the sine function is negative. Therefore:
$\sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}$

STEP 6

Find $\csc 300^\circ$ by taking the reciprocal of $\sin 300^\circ$ .
$\csc 300^\circ = \frac{1}{\sin 300^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}$

STEP 7

Choose the correct answer from the given options.
The correct answer is:
c) $-\frac{2 \sqrt{3}}{3}$

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QuestionQuestion 7 - 5 marks What is the exact value of $\csc 300^{\circ}$ ? a) $\frac{\sqrt{3}}{2}$ b) $\frac{2}{\sqrt{3}}$ c) $-\frac{2 \sqrt{3}}{3}$ d) $\frac{1}{2}$

Studdy Solution

STEP 1

1. The angle $300^\circ$ is measured in standard position.
2. The cosecant function, $\csc \theta$ , is the reciprocal of the sine function, $\sin \theta$ .

STEP 2

STEP 3

Determine the reference angle for $300^\circ$ .
The reference angle is calculated as:
$360^\circ - 300^\circ = 60^\circ$

STEP 4

Identify the quadrant in which $300^\circ$ lies.
Since $300^\circ$ is between $270^\circ$ and $360^\circ$ , it lies in the fourth quadrant.

STEP 5

Calculate $\sin 300^\circ$ using the reference angle and quadrant information.
In the fourth quadrant, the sine function is negative. Therefore:
$\sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}$

STEP 6

Find $\csc 300^\circ$ by taking the reciprocal of $\sin 300^\circ$ .
$\csc 300^\circ = \frac{1}{\sin 300^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}$

STEP 7

Choose the correct answer from the given options.
The correct answer is:
c) $-\frac{2 \sqrt{3}}{3}$

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QuestionQuestion 7 - 5 marks What is the exact value of csc⁡300∘\csc 300^{\circ}csc300∘ ? a) 32\frac{\sqrt{3}}{2}23​​ b) 23\frac{2}{\sqrt{3}}3​2​ c) −233-\frac{2 \sqrt{3}}{3}−323​​ d) 12\frac{1}{2}21​

Studdy Solution

STEP 1

1. The angle 300∘ 300^\circ 300∘ is measured in standard position.2. The cosecant function, csc⁡θ\csc \thetacscθ, is the reciprocal of the sine function, sin⁡θ\sin \thetasinθ.

STEP 2

STEP 3

Determine the reference angle for 300∘ 300^\circ 300∘.The reference angle is calculated as:360∘−300∘=60∘ 360^\circ - 300^\circ = 60^\circ 360∘−300∘=60∘

STEP 4

Identify the quadrant in which 300∘ 300^\circ 300∘ lies.Since 300∘ 300^\circ 300∘ is between 270∘ 270^\circ 270∘ and 360∘ 360^\circ 360∘, it lies in the fourth quadrant.

STEP 5

Calculate sin⁡300∘\sin 300^\circsin300∘ using the reference angle and quadrant information.In the fourth quadrant, the sine function is negative. Therefore:sin⁡300∘=−sin⁡60∘=−32 \sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2} sin300∘=−sin60∘=−23​​

STEP 6

Find csc⁡300∘\csc 300^\circcsc300∘ by taking the reciprocal of sin⁡300∘\sin 300^\circsin300∘.csc⁡300∘=1sin⁡300∘=1−32=−23 \csc 300^\circ = \frac{1}{\sin 300^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} csc300∘=sin300∘1​=−23​​1​=−3​2​

STEP 7

Choose the correct answer from the given options.The correct answer is:c) −233-\frac{2 \sqrt{3}}{3}−323​​

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QuestionQuestion 7 - 5 marks What is the exact value of $\csc 300^{\circ}$ ? a) $\frac{\sqrt{3}}{2}$ b) $\frac{2}{\sqrt{3}}$ c) $-\frac{2 \sqrt{3}}{3}$ d) $\frac{1}{2}$

1. The angle $300^\circ$ is measured in standard position.
2. The cosecant function, $\csc \theta$ , is the reciprocal of the sine function, $\sin \theta$ .

Determine the reference angle for $300^\circ$ .
The reference angle is calculated as:
$360^\circ - 300^\circ = 60^\circ$

Identify the quadrant in which $300^\circ$ lies.
Since $300^\circ$ is between $270^\circ$ and $360^\circ$ , it lies in the fourth quadrant.

Calculate $\sin 300^\circ$ using the reference angle and quadrant information.
In the fourth quadrant, the sine function is negative. Therefore:
$\sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}$

Find $\csc 300^\circ$ by taking the reciprocal of $\sin 300^\circ$ .
$\csc 300^\circ = \frac{1}{\sin 300^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}$

Choose the correct answer from the given options.
The correct answer is:
c) $-\frac{2 \sqrt{3}}{3}$