Given [tex]$\sin t=\frac{x}{x+1}$[/tex], find [tex]$\cos \left(\sin ^{-1}\left(\frac{x}{x+1}\right)\right)$[/tex]
Answer (2)
Let y = sin − 1 ( x + 1 x ) , so sin y = x + 1 x .
Use the identity sin 2 y + cos 2 y = 1 to find cos 2 y = 1 − sin 2 y = 1 − ( x + 1 x ) 2 .
Simplify the expression to get cos 2 y = ( x + 1 ) 2 2 x + 1 .
Take the square root to find cos y = x + 1 2 x + 1 .
x + 1 2 x + 1
Explanation
Problem Setup and Variable Assignment We are given that sin t = x + 1 x and we want to find cos ( sin − 1 ( x + 1 x )) . Let y = sin − 1 ( x + 1 x ) . Then sin y = x + 1 x . We want to find cos y .
Applying Trigonometric Identity We know the trigonometric identity sin 2 y + cos 2 y = 1 . Therefore, cos 2 y = 1 − sin 2 y . Substituting sin y = x + 1 x , we get cos 2 y = 1 − ( x + 1 x ) 2 .
Simplifying the Expression Now we simplify the expression: cos 2 y = 1 − ( x + 1 ) 2 x 2 = ( x + 1 ) 2 ( x + 1 ) 2 − x 2 = ( x + 1 ) 2 x 2 + 2 x + 1 − x 2 = ( x + 1 ) 2 2 x + 1 .
Taking the Square Root Taking the square root of both sides, we have cos y = ( x + 1 ) 2 2 x + 1 = ∣ x + 1∣ 2 x + 1 . Since the range of sin − 1 ( u ) is [ − 2 π , 2 π ] , cos ( sin − 1 ( u )) is always non-negative. Therefore, we need to ensure that the expression is non-negative.
Considering the Sign If 0"> x + 1 > 0 , then ∣ x + 1∣ = x + 1 , and cos y = x + 1 2 x + 1 . If x + 1 < 0 , then ∣ x + 1∣ = − ( x + 1 ) , and cos y = − x + 1 2 x + 1 . However, since cos y must be non-negative, we must have 0"> x + 1 > 0 , so -1"> x > − 1 . Also, 2 x + 1 must be non-negative, so 2 x + 1 ≥ 0 , which means x ≥ − 2 1 . Thus, x ≥ − 2 1 .
Final Answer Therefore, cos ( sin − 1 ( x + 1 x )) = x + 1 2 x + 1 .
Examples
Imagine you're designing a ramp for a skateboard park. The angle of the ramp is related to the height and length of the ramp. If you know the ratio of the height to the length (which is like the sine of the angle), you can use the inverse sine function to find the angle. Then, you might need to find the cosine of that angle to calculate other aspects of the ramp's design, such as the horizontal distance covered by the ramp. This problem demonstrates how trigonometric functions and their inverses are used in real-world applications like engineering and design.
To find cos ( sin − 1 ( x + 1 x )) , we defined y = sin − 1 ( x + 1 x ) and used the identity sin 2 y + cos 2 y = 1 . After simplifying, our answer is x + 1 2 x + 1 assuming 0"> x + 1 > 0 .
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