Which is a solution of [tex]$\tan x=\frac{\sqrt{3}}{3}$[/tex]? Give your answer in radians.
A. 0.65
B. [tex]$\frac{\pi}{6}$[/tex]
C. [tex]$\frac{\pi}{3}$[/tex]
D. 6.15
Answer (2)
The problem requires finding a solution to tan x = 3 3 from the given options.
Recall that tan ( 6 π ) = 3 3 .
Verify that the other options are not solutions by calculating their tangent values.
The solution is 6 π .
Explanation
Problem Analysis We are given the equation tan x = 3 3 and asked to find a solution in radians from the given options: 0.65, 6 π , 3 π , 6.15.
Recall Trigonometric Values We need to determine which of the given values for x satisfies the equation. We know that tan ( 6 π ) = c o s ( 6 π ) s i n ( 6 π ) = 2 3 2 1 = 3 1 = 3 3 .
Verify the Solution Therefore, x = 6 π is a solution to the equation tan x = 3 3 . Let's check the other options using a calculator: tan ( 0.65 ) ≈ 0.7602 tan ( 3 π ) = 3 ≈ 1.732 tan ( 6.15 ) ≈ − 0.134 None of these values equal 3 3 ≈ 0.577 .
Final Answer Thus, the solution from the given options is 6 π .
Examples
The tangent function is used in many areas of physics and engineering, such as calculating the angle of elevation in surveying or determining the slope of a line. For example, if you are building a ramp and need it to have a certain slope, you can use the tangent function to determine the angle of elevation needed for the ramp. If the ratio of the height to the horizontal distance is 3 3 , then the angle of elevation is 6 π radians or 30 degrees.
The solution to the equation tan x = 3 3 is 6 π . This is because tan ( 6 π ) = 3 3 , and the other options do not satisfy this equation. Therefore, the correct option is 6 π .
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