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Math  /  Trigonometry

Question2) Determine the exact value: (6 marks) csc⁡240∘\csc 240^{\circ} cot⁡120∘\cot 120^{\circ}

Studdy Solution

STEP 1

1. The problem involves trigonometric functions: cosecant (csc⁡\csc) and cotangent (cot⁡\cot).
2. We need to find the exact values of these trigonometric functions for the given angles.
3. We will use known identities and properties of trigonometric functions to find these values.

STEP 2

1. Determine the reference angle and quadrant for 240∘240^\circ and 120∘120^\circ.
2. Calculate csc⁡240∘\csc 240^\circ.
3. Calculate cot⁡120∘\cot 120^\circ.
4. Multiply the results to find the exact value.

STEP 3

Determine the reference angle and quadrant for each angle:
- 240∘240^\circ is in the third quadrant. The reference angle is 240∘−180∘=60∘240^\circ - 180^\circ = 60^\circ. - 120∘120^\circ is in the second quadrant. The reference angle is 180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ.

STEP 4

Calculate csc⁡240∘\csc 240^\circ:
- In the third quadrant, sine is negative. Therefore, sin⁡240∘=−sin⁡60∘=−32\sin 240^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}. - csc⁡240∘=1sin⁡240∘=1−32=−23\csc 240^\circ = \frac{1}{\sin 240^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}.
Rationalize the denominator:
- csc⁡240∘=−23×33=−233\csc 240^\circ = -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}.

STEP 5

Calculate cot⁡120∘\cot 120^\circ:
- In the second quadrant, tangent is negative. Therefore, tan⁡120∘=−tan⁡60∘=−3\tan 120^\circ = -\tan 60^\circ = -\sqrt{3}. - cot⁡120∘=1tan⁡120∘=1−3=−13\cot 120^\circ = \frac{1}{\tan 120^\circ} = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}}.
Rationalize the denominator:
- cot⁡120∘=−13×33=−33\cot 120^\circ = -\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}.

STEP 6

Multiply the results to find the exact value:
- csc⁡240∘⋅cot⁡120∘=(−233)⋅(−33)\csc 240^\circ \cdot \cot 120^\circ = \left(-\frac{2\sqrt{3}}{3}\right) \cdot \left(-\frac{\sqrt{3}}{3}\right).
Simplify the expression:
- csc⁡240∘⋅cot⁡120∘=23⋅39=2⋅39=69=23\csc 240^\circ \cdot \cot 120^\circ = \frac{2\sqrt{3} \cdot \sqrt{3}}{9} = \frac{2 \cdot 3}{9} = \frac{6}{9} = \frac{2}{3}.
The exact value is:
23 \boxed{\frac{2}{3}}

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