Module 6: Practice Problems
1
Solve 2sinθ+1=0 on [0,2π).
Solution
sinθ=−1/2. θ=7π/6, 11π/6.
2
Solve cos²θ−cosθ=0 on [0,2π).
Solution
cosθ(cosθ−1)=0. cos=0:π/2,3π/2. cos=1:0. {0,π/2,3π/2}.
3
Solve sin(2x)=1 on [0,2π).
Solution
2x=π/2,5π/2. x=π/4, 5π/4.
4
Find arcsin(√3/2).
Solution
π/3.
5
Find arccos(−1/2).
Solution
2π/3.
6
Solve 2cos²θ+3cosθ+1=0 on [0,2π).
Solution
(2cos+1)(cos+1)=0. cos=−1/2:2π/3,4π/3. cos=−1:π. {2π/3,π,4π/3}.
7
Find tan(arcsin(4/5)).
Solution
θ=arcsin(4/5), cosθ=3/5. tan=4/3.
8
Solve cos(3x)=−1 on [0,2π).
Solution
3x=π,3π,5π. x=π/3, π, 5π/3.
9
Write the general solution of cosθ=√2/2.
Solution
θ=π/4+2nπ or θ=7π/4+2nπ (equivalently ±π/4+2nπ).
10
Solve sin2θ=sinθ on [0,2π).
Solution
2sinθcosθ−sinθ=0, sinθ(2cosθ−1)=0. sin=0:0,π. cos=1/2:π/3,5π/3. {0,π/3,π,5π/3}.