Re: How to Solve...
cos3? = 1/?2 for 0 ? ? < 2?
hello, pigg0606,
You're on the right track with 3? = ?/4, 7?/4.
As you've noted, the constraint conditions allow for two solutions since cosine is positive only in the 1st and 4th quadrants.
Draw a triangle in the 1st quadrant, making the adjacent side 1 and the hypotenuse ?2 (since we’re evaluating a cosine expression). This triangle contains an angle equal to 3?. (A similar, inverted triangle exists in the 4th quadrant.)
Using the Pythagorean Theorem to solve for the third (opposite) side, we find it has a value of 1 also. This is a 45-45-90 right triangle.
Therefore, 3? = 45 degrees (or ?/4 radians – 1st quad) or 3? = 315 degrees (or 7?/4 radians – 4th quad). Divide by 3:
So ? = ?/12, and ? = 7?/12.
In the second part of the problem, we are not constrained by 0 ? ? < 2?. We will have an infinite number of solutions. We show this by adding the expression 2?n (where n is the set of integers) to each answer in part one.
? = ?/12 + 2?n
? = 7?/12 + 2?n
