Find the exact value of sin300 degrees Find the exact vale of sec330 degrees
The Correct Answer and Explanation is :
Let’s find the exact values of ( \sin 300^\circ ) and ( \sec 330^\circ ).
Step 1: Find ( \sin 300^\circ )
The angle 300° lies in the fourth quadrant of the unit circle.
- To find ( \sin 300^\circ ), we first note that 300° is ( 360^\circ – 60^\circ ), meaning it’s a reference angle of 60°.
- In the fourth quadrant, the sine function is negative.
- Therefore, ( \sin 300^\circ = -\sin 60^\circ ).
- We know from the unit circle that ( \sin 60^\circ = \frac{\sqrt{3}}{2} ).
Thus,
[
\sin 300^\circ = -\frac{\sqrt{3}}{2}.
]
Step 2: Find ( \sec 330^\circ )
The angle 330° lies in the fourth quadrant of the unit circle.
- To find ( \sec 330^\circ ), we first determine the cosine of 330°, because ( \sec \theta = \frac{1}{\cos \theta} ).
- 330° is ( 360^\circ – 30^\circ ), meaning the reference angle is 30°.
- In the fourth quadrant, the cosine function is positive.
- Therefore, ( \cos 330^\circ = \cos 30^\circ ).
- From the unit circle, we know that ( \cos 30^\circ = \frac{\sqrt{3}}{2} ).
Thus,
[
\sec 330^\circ = \frac{1}{\cos 330^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}.
]
To rationalize the denominator:
[
\sec 330^\circ = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}.
]
Final Answers:
- ( \sin 300^\circ = -\frac{\sqrt{3}}{2} )
- ( \sec 330^\circ = \frac{2\sqrt{3}}{3} )
Explanation:
Both of these values come from understanding the unit circle and the properties of trigonometric functions in different quadrants. The reference angle helps us identify the corresponding sine or cosine value for the standard angles (like 30°, 60°, etc.), and the signs of the functions in each quadrant help us determine the final answer.