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What trigonometric expression can be used to find the value of [tex]$x$[/tex]? Replace [tex]$a$[/tex] and [tex]$b$[/tex] with the correct values.
[tex]a \tan (b)[/tex]
Answer (2)
To find the angle x in a right triangle, use the tangent function defined as tan ( x ) = b a , where a is the length of the opposite side and b is the length of the adjacent side. The angle x can then be calculated using the inverse tangent function: x = tan − 1 ( b a ) .
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The tangent of angle x in a right triangle is defined as the ratio of the opposite side ( a ) to the adjacent side ( b ).
Express the relationship as tan ( x ) = b a .
Therefore, the trigonometric expression to find the value of x is tan ( x ) = b a .
Explanation
Problem Analysis We are given a right triangle with angle x , side a opposite to x , and side b adjacent to x . We need to find a trigonometric expression for x using a and b .
Tangent Function Definition Recall the definition of the tangent function: in a right triangle, the tangent of an angle is the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. Therefore, we have: tan ( x ) = adjacent opposite = b a
Final Expression Thus, the trigonometric expression to find the value of x is tan ( x ) = b a .
Examples
In surveying, if you know the height ( a ) of a building and the horizontal distance ( b ) to the building, you can use the tangent function to find the angle of elevation ( x ) from your position to the top of the building. This is useful for determining the height of objects or the distance to them using angles and trigonometry.
