[tex]\sin \left(\frac{3 \pi}{4}\right)=[/tex]
A. [tex]$\frac{\sqrt{3}}{2}$[/tex]
B. [tex]$\frac{\sqrt{2}}{2}$[/tex]
C. [tex]$\frac{1}{2}$[/tex]
D. [tex]$-\frac{\sqrt{2}}{2}$[/tex]
Answer (2)
Recognize that 4 3 π lies in the second quadrant.
Express 4 3 π as π − 4 π .
Apply the identity sin ( π − x ) = sin ( x ) to get sin ( 4 3 π ) = sin ( 4 π ) .
Evaluate sin ( 4 π ) to find the answer: 2 2 .
Explanation
Problem Analysis We are asked to find the value of sin ( 4 3 π ) . This involves understanding trigonometric functions and their values at specific angles.
Angle Location The angle 4 3 π is in the second quadrant of the unit circle. In the second quadrant, the sine function is positive. We can express 4 3 π as π − 4 π .
Applying Trigonometric Identity Using the identity sin ( π − x ) = sin ( x ) , we have sin ( 4 3 π ) = sin ( π − 4 π ) = sin ( 4 π ) .
Evaluating Sine We know that sin ( 4 π ) = 2 2 . Therefore, sin ( 4 3 π ) = 2 2 .
Final Answer The value of sin ( 4 3 π ) is 2 2 .
Examples
Understanding trigonometric functions like sine is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a pendulum, the sine function describes the displacement of the pendulum bob from its equilibrium position over time. Similarly, in electrical engineering, alternating current (AC) waveforms are often modeled using sinusoidal functions, where the sine function helps describe the voltage or current as it varies with time. Knowing the values of sine at common angles like 4 3 π allows engineers and physicists to quickly estimate and analyze these systems.
The value of sin ( 4 3 π ) is 2 2 , as determined by using trigonometric identities. This angle is located in the second quadrant where sine is positive. Therefore, the correct choice is option B: 2 2 .
;
