Write one trigonometric equation that can be used to find the value of $x$ by replacing the variable $a$ with the correct value.
$x=\sin ^{-1}(a)$
Answer (1)
Substitute a = 2 1 into the equation x = sin − 1 ( a ) .
Obtain the equation x = sin − 1 ( 2 1 ) .
Solve for x , knowing that sin − 1 ( 2 1 ) = 6 π .
The final answer is x = 6 π .
Explanation
Understanding the Problem The problem provides the equation x = sin − 1 ( a ) and asks us to replace the variable a with a specific value to create a trigonometric equation that can be used to find the value of x . The domain of the inverse sine function, sin − 1 ( a ) , is − 1 ≤ a ≤ 1 . Therefore, we need to choose a value for a within this interval.
Substituting a Value for a Let's choose a = 2 1 . This value is within the domain of the inverse sine function. Substituting this value into the given equation, we get x = sin − 1 ( 2 1 ) .
Solving for x The equation x = sin − 1 ( 2 1 ) is a valid trigonometric equation. We know that sin ( 6 π ) = 2 1 , so sin − 1 ( 2 1 ) = 6 π . Therefore, x = 6 π .
Final Answer Thus, the trigonometric equation x = sin − 1 ( 2 1 ) can be used to find the value of x , which is 6 π .
Examples
Trigonometric equations are used in various fields such as physics, engineering, and navigation. For example, if you are designing a suspension bridge, you need to calculate the angles and tensions in the cables. These calculations often involve trigonometric functions and their inverses. Similarly, in navigation, determining the position of a ship or aircraft relies on solving trigonometric equations based on angles of elevation or depression to known landmarks or celestial bodies. Understanding how to manipulate and solve these equations is crucial for accurate and safe designs and calculations.
