Question

Solve the following trigonometric equations when 0⁰ <=ᶿ <= 90⁰
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1. Cos ²ᶿ - 3 cos ᶿ + 2 = 2 sin ² ᶿ 0⁰ <ᶿ≤ 90⁰

Answer 1

$$\begin{gathered} {\cos }^2\mathrm{\theta }-3\cos \mathrm{\theta }+2=2{\sin }^2\mathrm{\theta } \\ \Rightarrow {\cos }^2\mathrm{\theta }-3\cos \mathrm{\theta }+2=2\left(1-{\cos }^2\mathrm{\theta }\right) \\ \Rightarrow {\cos }^2\mathrm{\theta }-3\cos \mathrm{\theta }+2=2-2{\cos }^2\mathrm{\theta } \\ \Rightarrow 3{\cos }^2\mathrm{\theta }-3\cos \mathrm{\theta }=0 \\ \Rightarrow 3\cos \mathrm{\theta }\left(\cos \mathrm{\theta }-1\right)=0 \\ \Rightarrow \cos \mathrm{\theta }\left(\cos \mathrm{\theta }-1\right)=0 \\ \Rightarrow \cos \mathrm{\theta }=0\mathrm{or}\cos \mathrm{\theta }-1=0 \\ \Rightarrow \cos \mathrm{\theta }=0\mathrm{or}\cos \mathrm{\theta }=1 \\ \Rightarrow \cos \mathrm{\theta }=\cos 90°\mathrm{or}\cos \mathrm{\theta }=\cos 0° \\ \Rightarrow \mathrm{\theta }=90°\mathrm{or}\mathrm{\theta }=0° \end{gathered}$$