Section 1.4 : Solving Trig Equations

Without using a calculator find the solution(s) to the following equations. If an interval is given find only those solutions that are in the interval. If no interval is given find all solutions to the equation.

1. \(4\sin \left( {3t} \right) = 2\) Solution
2. \(4\sin \left( {3t} \right) = 2\) in \(\displaystyle \left[ {0,\frac{{4\pi }}{3}} \right]\) Solution
3. \(\displaystyle 2\cos \left( {\frac{x}{3}} \right) + \sqrt 2 = 0\) Solution
4. \(\displaystyle 2\cos \left( {\frac{x}{3}} \right) + \sqrt 2 = 0\) in \(\left[ { - 7\pi ,7\pi } \right]\) Solution
5. \(4\cos \left( {6z} \right) = \sqrt {12} \) in \(\displaystyle \left[ {0,\frac{\pi }{2}} \right]\) Solution
6. \(\displaystyle 2\sin \left( {\frac{{3y}}{2}} \right) + \sqrt 3 = 0\) in \(\displaystyle \left[ { - \frac{{7\pi }}{3},0} \right]\) Solution
7. \(8\tan \left( {2x} \right) - 5 = 3\) in \(\displaystyle \left[ { - \frac{\pi }{2},\frac{{3\pi }}{2}} \right]\) Solution
8. \(16 = - 9\sin \left( {7x} \right) - 4\) in \(\displaystyle \left[ { - 2\pi ,\frac{{9\pi }}{4}} \right]\) Solution
9. \(\displaystyle \sqrt 3 \tan \left( {\frac{t}{4}} \right) + 5 = 4\) in \(\left[ {0,4\pi } \right]\) Solution
10. \(\sqrt 3 \csc \left( {9z} \right) - 7 = - 5\) in \(\displaystyle \left[ { - \frac{\pi }{3},\frac{{4\pi }}{9}} \right]\) Solution
11. \(\displaystyle 1 - 14\cos \left( {\frac{{2x}}{5}} \right) = - 6\) in \(\displaystyle \left[ {5\pi ,\frac{{40\pi }}{3}} \right]\) Solution
12. \(\displaystyle 15 = 17 + 4\cos \left( {\frac{y}{7}} \right)\) in \(\left[ {10\pi ,15\pi } \right]\) Solution