# Trigonometric Equation – Solve for theta

By | August 17, 2016

## Trigonometric equation where you solve for theta

In the following tutorial the student will learn how to solve a trigonometric equation and solve for the possible values of Θ over the specified interval.  The trigonometric equation is solved in the tutorial, as well as, in the attached document.

`(tan²Θ)(sinΘ) = 0`

Notice that both tan²(Θ) and sin(Θ) are factors and that their product is 0.

This is exactly what we need to apply the Zero Product Property.

`If xy = 0 then x = 0 or y = 0.`

In common language it means that if either x or y is 0 then the result is going to always be zero.  We use this to our advantage by setting each factor equal to zero and then solving.

`tan²(Θ)=0 and sin(Θ)=0`

Now we fall back to what we know about the basic trigonometric functions and the unit circle.

`tan(Θ) = y/x and sin(Θ) = y/r`

First let’s talk about tangent and the ratio y/x.  When Θ is 0 radians and π radians, y is 0 and the ratio evaluates to 0.

Now let’s talk about sine and the ratio y/r.  In this case we get the same results as tangent.  When Θ is 0 radians or π radians, y is 0 and the ratio again evaluates to 0.

Note that this equation actually has an infinite number of solutions but is limit the interval to [0,2π).  The interval limits the solutions to Θ = { 0, π }.