Answer:
[tex]\theta_{1} \approx 0.379\cdot \pi[/tex] or [tex]\theta_{1} \approx 1.379\cdot \pi[/tex], [tex]\theta_{2} = 0.25\cdot \pi[/tex] or [tex]\theta_{2} = 1.25\cdot \pi[/tex]
Step-by-step explanation:
Let use the following substitution formula:
[tex]u = \tan \theta[/tex]
The trigonometric expression is converted into an algebraic one, a second-order polynomial:
[tex]2\cdot u^{2}-7\cdot u + 5 = 0[/tex]
Roots can be found by using the General Equation for Second-Order Polynomial:
[tex]u = \frac{7\pm \sqrt{49 - 40} }{4}[/tex]
Roots are [tex]u_{1} = 2.5[/tex] and [tex]u_{2} = 1[/tex]. As tangent function has a periodicity of [tex]\pi[/tex], solutions of [tex]u_{1}[/tex] and [tex]u_{2}[/tex] belong to first and third quadrants. Then, angles can be easily found by using inverse trigonometric functions:
[tex]\theta_{1} = \tan^{-1} u_{1}[/tex]
[tex]\theta_{1} \approx 0.379\cdot \pi[/tex] or [tex]\theta_{1} \approx 1.379\cdot \pi[/tex]
[tex]\theta_{2} = \tan^{-1} u_{2}[/tex]
[tex]\theta_{2} = 0.25\cdot \pi[/tex] or [tex]\theta_{2} = 1.25\cdot \pi[/tex]
