# Video: Solving Trigonometric Equations Involving Special Angles

If π β (180Β°, 360Β°) and sin π + cos π = β1, find the value of π.

03:49

### Video Transcript

If π exists between 180 degrees and 360 degrees and sin π plus cos π equals negative one, find the value of π.

Letβs first consider the equation sin π plus cos π equals negative one. Our first step here is to square both sides of the equation. This gives us sin π plus cos π all squared is equal to negative one squared. In order to square sin π plus cos π, we need to write the brackets or parentheses out twice.

Negative one squared is equal to one, as multiplying a negative number by another negative gives us a positive. We can expand or distribute the parentheses using the FOIL method. Multiplying the first terms, sin π and sin π, gives us sin squared π. Multiplying the outside terms gives us sin π cos π. Multiplying the inside terms also gives us sin π cos π. And finally, multiplying the last terms gives us cos squared π.

This leaves us with the equation sin squared π plus sin π cos π plus sin π cos π plus cos squared π is equal to one. The middle two terms are the same, so they can be grouped or collected. This leaves us with two sin π cos π. We can also drop down the other three terms, sin squared π, cos squared π, and one.

One of our trigonometrical identities states that sin squared π plus cos squared π is equal to one. This means that the equation could be rewritten as two sin π cos π plus one equals one. Another one of our identities, one of the double-angle formulae, states that two sin π cos π is equal to sin two π. This means that sin two π plus one is equal to one. We can subtract one from both sides of this equation. This means that sin two π is equal to zero.

We need to solve this equation for all values of π between 180 degrees and 360 degrees, exclusive. Not including 180 or 360. If we consider the graph of sin two π, it has a maximum value of one and a minimum value of negative one. The values of π that give us these maximum and minimum values are shown on the graph. Sketching this graph shows us that there are numerous values where sin two π is equal to zero. They occur at zero degrees, 90 degrees, 180 degrees, 270 degrees, and 360 degrees. However, since weβre looking for values between 180 and 360 degrees, the only solution in this case is π equals 270 degrees. If sin π plus cos π is equal to negative one, the solution between 180 and 360 degrees is 270 degrees.