Type the correct answer in the box. Write one trigonometric equation that can be used to find the value of [tex]$x$[/tex] by replacing the variable [tex]$a$[/tex] with the correct value. [tex]$x=\sin ^{-1}(3 / 5)$[/tex]
Answer (2)
Apply the sine function to both sides of the equation x = sin − 1 ( 5 3 ) .
Simplify the resulting equation using the property sin ( sin − 1 ( y )) = y .
Obtain the trigonometric equation sin ( x ) = 5 3 .
The final trigonometric equation is: sin ( x ) = 5 3
Explanation
Analyze the given equation. We are given that x = sin − 1 ( 5 3 ) . Our goal is to find a trigonometric equation that expresses this relationship.
Apply sine function to both sides. To find the trigonometric equation, we can apply the sine function to both sides of the equation x = sin − 1 ( 5 3 ) . This gives us sin ( x ) = sin ( sin − 1 ( 5 3 )) .
Simplify the equation. Since sin ( sin − 1 ( y )) = y for y in the domain of sin − 1 , we can simplify the right side of the equation: sin ( x ) = 5 3 .
State the trigonometric equation. Therefore, the trigonometric equation we are looking for is sin ( x ) = 5 3 .
Examples
Imagine you are designing a ramp for a skateboard park. The angle of the ramp, x , is such that the ratio of the height of the ramp to its length is 5 3 . This means sin ( x ) = 5 3 , where x is the angle of elevation of the ramp. This equation helps you determine the angle needed for the ramp to meet safety and performance standards.
The trigonometric equation derived from x = sin − 1 ( 5 3 ) is sin ( x ) = 5 3 . This is achieved by applying the sine function to both sides of the original equation. Therefore, this relationship expresses how the sine of the angle x is equal to 5 3 .
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