Find the exact value of sin 330°.
Answer (2)
Recognize that 33 0 ∘ is equivalent to 36 0 ∘ − 3 0 ∘ .
Use the property that sin ( 36 0 ∘ − x ) = − sin ( x ) .
Find the value of sin ( 3 0 ∘ ) , which is 2 1 .
Apply the negative sign to find the value of sin ( 33 0 ∘ ) : − 2 1 .
Explanation
Problem Analysis We are asked to find the exact value of sin ( 33 0 ∘ ) .
Expressing the angle First, we recognize that 33 0 ∘ is in the fourth quadrant. We can express 33 0 ∘ as 36 0 ∘ − 3 0 ∘ .
Applying the sine property Using the property that sin ( 36 0 ∘ − x ) = − sin ( x ) , we have sin ( 33 0 ∘ ) = sin ( 36 0 ∘ − 3 0 ∘ ) = − sin ( 3 0 ∘ )
Finding the value We know that sin ( 3 0 ∘ ) = 2 1 . Therefore, sin ( 33 0 ∘ ) = − sin ( 3 0 ∘ ) = − 2 1
Examples
Understanding trigonometric functions like sine is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum, the sine function helps describe the pendulum's displacement from its resting position over time. Similarly, in electrical engineering, sine waves are used to model alternating current (AC) circuits. Knowing the values of sine for common angles allows engineers to predict and control the behavior of these systems.
The exact value of sin ( 33 0 ∘ ) is − 2 1 . This is derived from recognizing that 33 0 ∘ is equivalent to 36 0 ∘ − 3 0 ∘ , leading to the application of the sine property. The sine of 3 0 ∘ is 2 1 , which yields a negative value in the fourth quadrant.
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