Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.)
tan θ = −square root of 3
If you are studying Calculus, you should make an effort to memorize the ratios of two important triangles:
1. the 45-45-90 degree triangle, with sides
1 : 1 : √2 corresponding to the angles
2. the 30-60-90 degree triangle, with
sides
1 : √3 : 2
so in this case
if tan θ = +√3 then θ = 60°
but the tangent is negative in quadrants II or IV
so θ = 120° or θ = 300°
you should also know that
120° = 2π/3, and 300° = 5π/3
You did not state if you want the anser in degrees or in radians.
To solve the equation tan θ = −√3, we can use the inverse tangent function (also known as arctan or tan^(-1)) to find the value of θ.
Step 1: Take the inverse tangent (arctan or tan^(-1)) of both sides of the equation:
arctan(tan θ) = arctan(−√3)
Step 2: Since the tangent function has a period of π (180 degrees), we can add or subtract π from the result obtained in step 1 to find other solutions. In this case, we will use the principal value, which is between -π/2 and π/2.
θ = arctan(−√3)
Using a calculator or any online trigonometric calculator, we can find the value of arctan(−√3) approximately equal to -60 degrees (-π/3 radians) or 300 degrees (5π/3 radians).
Therefore, the solutions for θ are -60 degrees (-π/3 radians) or 300 degrees (5π/3 radians).
As a comma-separated list, the solutions would be: -60°, 300° or approximately -1.05 radians, 5.24 radians.