04.23.2019: One Lesson of Math - Trigonometric Equations and Identities, 2/6: Solving Trigonometric

Today's soundtrack is Petra: Petra Means Rock, a compilation album that includes many of the hits from their earlier albums.

This afternoon, I'm learning how to solve trig equations without using a calculator.

Trigonometric equations include a trig function and a variable - for example, y = sin x.

How to solve first-degree trigonometric equations for exact values

  1. Isolate the trigonometric function

  2. Using absolute values of the equation, refer to the special triangles and find the matching function and ratio to determine the reference angle, a

  3. Based on whether the original equation was positive or negative, use the unit circle to determine which quadrant(s) the original equation was situated in

  4. Based on the quadrants determined in Step 2, calculate a in standard position

  5. Give the general solution, which comes in the form [standard position angle] + n[period], n∈Z

We'll also be given trigonometric equations that must be factored before solving. We can factor trigonometric equations in the same ways that we solve other polynomials - through finding common factors, finding difference of squares, decomposition of factors, etc.

After we factor, we solve each factor for 0 on its own; at the end, we find the applicable solution or solutions. Below is an example of a trigonometric equation of the "messy trinomial" type.