Answer:
1. b) 135°, 225°
2. c) 0°, 180°
3. a) π/2
4. b) 30°, 150°, 210°, 330°
5. a) 1.25 radians and 4.39 radians
Step-by-step explanation:
Question 1
[tex]\begin{aligned}\cos \theta & = -\dfrac{\sqrt{2}}{2}\\\implies \theta & = \cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)\\\\& = 135^{\circ} \pm 360^{\circ}n, 225^{\circ} \pm 360^{\circ}\\\\& = 135^{\circ}, 225^{\circ}\quad \textsf{for}\:0^{\circ}\leq \theta < 360^{\circ}\end{aligned}[/tex]
Question 2
[tex]\begin{aligned}\cos^2 \theta-1 & = 0\\\implies \cos^2 \theta & = 1\\\cos \theta & = \pm1\\\theta & = \cos^{-1}(\pm1)\\& = 0^{\circ} \pm 360^{\circ}n, 180^{\circ} \pm360^{\circ}n\\& = 0^{\circ}, 180^{\circ}\quad \textsf{for}\:0^{\circ}\leq \theta < 360^{\circ}\end{aligned}[/tex]
Question 3
[tex]\begin{aligned}\sin \theta & = 1\\\implies \theta & = \sin^{-1}(1)\\\theta & = \dfrac{\pi}{2}\pm2\pi n\\\theta & = \dfrac{\pi}{2}\quad \textsf{for}\:0 \leq \theta < 2 \pi \end{aligned}[/tex]
Question 4
[tex]\begin{aligned}3 \tan^2 \theta-1 & = 0\\\implies \tan^2 \theta & = \dfrac{1}{3}\\\tan \theta & = \pm\dfrac{1}{\sqrt{3}}\\\theta & = \tan^{-1}\left(\pm\dfrac{1}{\sqrt{3}}\right)\\& = 30^{\circ}\pm180^{\circ}n, 150^{\circ}\pm180^{\circ}n\\& = 30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}\quad \textsf{for}\:0^{\circ}\leq \theta < 360^{\circ}\end{aligned}[/tex]
Question 5
[tex]\begin{aligned}2 \tan \theta -6 & = 0\\\implies \tan \theta & = 3\\\theta & = \tan^{-1}(3)\\\theta & = 1.25\pm \pi n\\\theta & = 1.25, 4.39 \quad \textsf{for}\:0^{\circ}\leq \theta < 2 \pi \end{aligned}[/tex]
