Video: Solving Trigonometric Equations Involving Special Angles

If πœƒ ∈ (180Β°, 360Β°) and sin πœƒ + cos πœƒ = βˆ’1, find the value of πœƒ.

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Video Transcript

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

Let’s first consider the equation sin πœƒ plus cos πœƒ equals negative one. Our first step here is to square both sides of the equation. This gives us sin πœƒ plus cos πœƒ all squared is equal to negative one squared. In order to square sin πœƒ plus cos πœƒ, 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œƒ and sin πœƒ, gives us sin squared πœƒ. Multiplying the outside terms gives us sin πœƒ cos πœƒ. Multiplying the inside terms also gives us sin πœƒ cos πœƒ. And finally, multiplying the last terms gives us cos squared πœƒ.

This leaves us with the equation sin squared πœƒ plus sin πœƒ cos πœƒ plus sin πœƒ cos πœƒ plus cos squared πœƒ 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œƒ cos πœƒ. We can also drop down the other three terms, sin squared πœƒ, cos squared πœƒ, and one.

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

We need to solve this equation for all values of πœƒ between 180 degrees and 360 degrees, exclusive. Not including 180 or 360. If we consider the graph of sin two πœƒ, it has a maximum value of one and a minimum value of negative one. The values of πœƒ 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œƒ is equal to zero. They occur at zero degrees, 90 degrees, 180 degrees, 270 degrees, and 360 degrees. However, since we’re looking for values between 180 and 360 degrees, the only solution in this case is πœƒ equals 270 degrees. If sin πœƒ plus cos πœƒ is equal to negative one, the solution between 180 and 360 degrees is 270 degrees.

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