Find the exact value of each trigonometric function sin 300
The Correct Answer and Explanation is:
To find the exact value of sin(300∘)\sin(300^\circ)sin(300∘), we need to first understand where 300∘300^\circ300∘ lies on the unit circle.
Step 1: Determine the reference angle
- The angle 300∘300^\circ300∘ is in the fourth quadrant of the unit circle.
- To find the reference angle, subtract 300∘300^\circ300∘ from 360∘360^\circ360∘: 360∘−300∘=60∘360^\circ – 300^\circ = 60^\circ360∘−300∘=60∘
Thus, the reference angle for 300∘300^\circ300∘ is 60∘60^\circ60∘.
Step 2: Determine the sign in the fourth quadrant
- In the fourth quadrant, the sine function is negative because the yyy-coordinate of points on the unit circle is negative in this quadrant.
Step 3: Use the known value of sine for the reference angle
- We know that sin(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}sin(60∘)=23.
Step 4: Apply the sign
Since 300∘300^\circ300∘ is in the fourth quadrant, and sine is negative in the fourth quadrant, the value of sin(300∘)\sin(300^\circ)sin(300∘) is: sin(300∘)=−sin(60∘)=−32\sin(300^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}sin(300∘)=−sin(60∘)=−23
Final Answer:
The exact value of sin(300∘)\sin(300^\circ)sin(300∘) is −32-\frac{\sqrt{3}}{2}−23.
