Sum and Difference Formulas for Sine in Radian Measure

In this math example, we are trying to use a sum or difference trig formula to calculate the exact value for angle in radian measure. We want to use the most common radian angles pi/6, pi/4, and pi/3 to do this calculation because these are known ratios. Each of these examples have given radian angles that have denominators of 12. Because of this, we first rewrite pi/6 as 2pi/12, pi/4 as 3pi/12, and pi/3 as 4pi/12. This makes the process of identifying which angles add or subtract to make the given angle easier.
sin(5pi/12)
The first example can be completed by using two angles that add together. We use the sum formula for sine to fill into the formula. I try to show where each angle is substituted into the formula by using color coding. After the problem is rewritten using our common angles, we fill in with the appropriate ratios to evaluate each trig function (sine and cosine). The answer is then simplified completely.
sin(13pi/12)
For the second problem, we are given an angle larger than pi. Because of this larger angle, no two pairs of our common angles add together to our given angle. Therefore, we want to find a reference angle. This means placing the given angle into the correct quadrant, drawing in our reference angle (always to the x-axis), and calculating the reference angle by subtracting the smaller angle from the larger that make up the reference angle. When we replace the original angle with the reference angle, we also must check to see the overall sign (positive/negative) of the function. I like to use the phrase "All Students Take Calculus" to remember in which quadrant each function is positive. This leads us to include a negative in front of our function that carries along when we fill into the difference formula for sine. The angles that are filled in for each piece of our formula are color coded to help see where they were filled in. After using the difference formula, we evaluate each of the known trig functions to give us nice exact ratios. The solution is then reduced fully to give the best solution possible.
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...