Solving a Trig Equation using Pythagorean Identity

In this math example, we find all solutions to a trigonometric equation on the interval from 0 to 2pi. This problem begins with both sine and cosine as part of the equation. Our first step is to replace sin^2(theta) with 1-cos^2(theta) by using the Pythagorean identity. Next, we work to get all of our terms on one side of the equation and zero on the other. We go ahead a rewrite our equation so that it looks more like something that we have solved before. To do so, we replace each cos(theta) with an x by using a "let statement". We then factor the resulting equation by using the AC method of factoring, set each factor equal to zero, and solve for x. Then x is replaced with cos(theta).
One of the resulting equations gives us cos(theta) = -1. For this ratio, we think about terminal points on the unit circle and think about what angle will create an x-value of -1.
The other equation gives cos(theta) = -1/2. This is a more difficult equation to solve that if we had sine or tangent equal to a negative ratio because we need to use the appropriate reference angle and place it in the correct quadrants to make sure that cosine will be negative. I like to utilize the phrase "All Students Take Calculus" to help remember the quadrant that each trig function will be positive (and negative). Cosine will be negative in the 2nd and 3rd quadrants so we take our time and use the reference angle to get our correct angles.
2cos(theta)+3=2sin^2(theta)
2cos(x)+3=sin^2(x)
This video contains examples that are from Algebra and Trigonometry, 1st ed, by Abramson, Belloit, Falduto, Gross, Lippman et al. It is an open-source textbook from OpenStax that you may download for
free at openstax.org/d.... The text is licensed under the Creative Commons Attribution license. creativecommon...