Which is equivalent to [tex]$\sin ^{-1}\left(\tan \left(\frac{\pi}{4}\right)\right)$[/tex] ? Give your answer in radians.
A. 0
B. [tex]$\frac{\pi}{2}$[/tex]
C. [tex]$\pi$[/tex]
D. [tex]$2 \pi$[/tex]
Answer (2)
Evaluate tan ( 4 π ) which equals 1.
Evaluate sin − 1 ( 1 ) .
sin − 1 ( 1 ) = 2 π .
The answer is 2 π .
Explanation
Understanding the Problem We are asked to find the value of sin − 1 ( tan ( 4 π )) . The answer should be in radians.
Evaluating the Tangent First, we need to evaluate tan ( 4 π ) . Recall that tan ( θ ) = c o s ( θ ) s i n ( θ ) . We know that sin ( 4 π ) = 2 2 and cos ( 4 π ) = 2 2 . Therefore, tan ( 4 π ) = 2 2 2 2 = 1 .
Evaluating the Inverse Sine Now we need to evaluate sin − 1 ( 1 ) . This is asking us, 'What angle has a sine of 1?' We know that sin ( 2 π ) = 1 . Therefore, sin − 1 ( 1 ) = 2 π .
Final Answer Thus, sin − 1 ( tan ( 4 π )) = sin − 1 ( 1 ) = 2 π .
Examples
Imagine you're designing a ramp and need to calculate the angle of elevation. This problem uses trigonometric functions to find angles based on known ratios. Understanding inverse trigonometric functions is crucial in fields like engineering, physics, and architecture, where angles and spatial relationships are essential for design and construction.
The expression sin − 1 ( tan ( 4 π ) ) evaluates to 2 π after calculating that tan ( 4 π ) = 1 and that sin − 1 ( 1 ) = 2 π . Thus, the answer is option B: 2 π .
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