Tan(x)=2, What is x in Degrees?

Tan(x)=2, What is x in Degrees?

To find the angle ‘x’ in degrees when tan(x) = 2, you can use the arctan (inverse tangent) function.

x = arctan(2) ≈ 63.43 degrees (rounded to two decimal places).

So, the angle ‘x’ is approximately 63.43 degrees when tan(x) = 2.

Understanding the Tangent Function

The tangent function in trigonometry gives the ratio between the side opposite an angle and the side adjacent to the angle in a right triangle. It is denoted as:

tan(θ) = Opposite / Adjacent

Where θ is the angle measure in degrees or radians. The tangent increases as the angle increases in a nonlinear fashion across the four quadrants.

When we are given tan(x) = 2, this means the tangent ratio for the angle x is 2.

Converting Between Radians and Degrees

Trigonometric functions like tangent can be defined with angle measures in either degrees or radians. A full circle is 360° or 2π radians.

To convert between degrees and radians:

Radians = (Degrees/360) * 2π Degrees = (Radians/2π) * 360

So if we want to find x in degrees when tan(x) = 2, we first need to determine x in radians using the tangent ratio, then convert from radians to degrees.

Using a Calculator

An easy way to find the angle in radians is to use a scientific calculator. With tan(x) = 2 entered, press the “inverse tan” or “arctan” button. This will give the angle measure as 1.107 radians.

To convert, use the formula:

Degrees = (Radians/2π) * 360

1.107 radians * (360/(2*π)) ≈ 63.4°

Therefore, if tan(x) = 2, the angle x is approximately 63.4 degrees.

Going Through the Unit Circle

We can also find the reference angle using a unit circle divided into quadrants with trig ratios. Looking at tan = 2, this matches the ratio at a 45° angle in Quadrant I.

However, for tan(x) = 2, x could be in Quadrant II instead of I because tan is positive in both quadrants. An angle of 180° – 45° = 135° in Quadrant II has tan(135) = 2.

So without a calculator, we can determine through the unit circle that if tan(x) = 2, x must be either 45° or 135° in degrees. Using a calculator confirms 135° is the acute angle measure.

In summary, finding the degree measure for an angle given its trig ratio involves knowing quadrant locations on the unit circle and converting between radians and degrees. Calculators make this process easier by providing the radian value first.