If [tex]\theta=\frac{\pi}{6}[/tex], find the exact value of each expression below.
(a) [tex]$\sin (-\theta)=$[/tex] [ ]
(b) [tex]$\sin ^2 \theta=$[/tex] [ ]
(c) [tex]$\sin 2 \theta=$[/tex] [ ]
Answer (1)
Use the property sin ( − x ) = − sin ( x ) to find sin ( − θ ) = − 2 1 .
Square the value of sin ( θ ) to find sin 2 ( θ ) = 4 1 .
Use the identity sin ( 2 x ) = 2 sin ( x ) cos ( x ) to find sin ( 2 θ ) = 2 3 .
The exact values are − 2 1 , 4 1 , and 2 3 .
Explanation
Problem Analysis We are given that θ = 6 π and we need to find the exact values of (a) sin ( − θ ) , (b) sin 2 ( θ ) , and (c) sin ( 2 θ ) .
Finding sin(-theta) (a) To find sin ( − θ ) , we use the property that sin ( − x ) = − sin ( x ) . Therefore, sin ( − θ ) = − sin ( 6 π ) . We know that sin ( 6 π ) = 2 1 . Thus, sin ( − θ ) = − 2 1 .
Finding sin^2(theta) (b) To find sin 2 ( θ ) , we need to square the value of sin ( θ ) . We know that sin ( 6 π ) = 2 1 . Therefore, sin 2 ( θ ) = ( 2 1 ) 2 = 4 1 .
Finding sin(2theta) (c) To find sin ( 2 θ ) , we can use the double angle identity sin ( 2 x ) = 2 sin ( x ) cos ( x ) . So, sin ( 2 θ ) = 2 sin ( 6 π ) cos ( 6 π ) . We know that sin ( 6 π ) = 2 1 and cos ( 6 π ) = 2 3 . Therefore, sin ( 2 θ ) = 2 × 2 1 × 2 3 = 2 3 .
Final Answer Therefore, the exact values are: (a) sin ( − θ ) = − 2 1 (b) sin 2 ( θ ) = 4 1 (c) sin ( 2 θ ) = 2 3
Examples
Understanding trigonometric functions like sine is crucial in many fields. For example, in physics, when analyzing simple harmonic motion, the sine function describes the position of an object oscillating over time. If you're designing a swing, knowing the sine of the angle helps determine the swing's height at any given point. Similarly, in electrical engineering, alternating current (AC) is modeled using sine waves, where understanding the sine function helps in analyzing voltage and current variations.
