Find the two standard angles between 0° and 360°

Question

Find the two standard angles between 0° and 360° whose tangent value is 10.

Round answer to 1 decimal place.

Answer 1

If  tan (θ) = 10    then,  θ - atan(10) = 84.3º

 

This solution corresponds to an angle θ in the first quadrant  (between 0° - 90°).

 

But, from the definition of tangent:

 

tan(θ) = sin(θ)/cos(θ)

 

The only two quadrants where tan(θ) can be positive are:

 

First quadrant  (0°  - 90°)    Sin(θ) positive

                                         Cos(θ) positive 

 

Third quadrant (180° - 270°)   Sin(θ) negative

                                             Cos(θ) negative

 

Then, the second angle Ψ such as tan(Ψ) = 10 is

Ψ = 180° + 84.3° = 264.3°

Answer 2

Given tan(θ) = 10.

 

Since the value of tan(θ) > 0, there is an angle for this tangent in the

first and third quadrants.

 

The simplest way to express the angle is to use the arc-tangent function.

This function should be on a graphing calculator.

 

In the first quadrant, θ = tan^-1(10) = 84,3°

In the third quadrant, θ = 180 + tan-1(10) = 264.3°