7.07 Solving trigonometric equations 2
Consider the equation \cos \left(\theta + 20 \degree\right) = \dfrac{1}{\sqrt{2}} for 0 \degree \leq \theta \leq 360 \degree.
If \alpha = \theta + 20 \degree, find the solutions of \cos \alpha = \dfrac{1}{\sqrt{2}} for 20 \degree \leq \alpha \leq 380 \degree.
Hence, solve \cos \left(\theta + 20 \degree\right) = \dfrac{1}{\sqrt{2}}.
Consider the equation \cos \left(\theta - 100 \degree\right) = \dfrac{1}{2} for 0 \degree \leq \theta \leq 360 \degree.
If \alpha = \theta - 100 \degree, find the solutions of \cos \alpha = \dfrac{1}{2} for - 100 \degree \leq \alpha \leq 260 \degree.
Hence, solve \cos \left(\theta - 100 \degree\right) = \dfrac{1}{2}.
Consider the equation \sin \left(\theta + 35 \degree\right) = \dfrac{1}{2} for 0 \degree \leq \theta \leq 360 \degree.
If \alpha = \theta + 35 \degree, find the solutions of \sin \alpha = \dfrac{1}{2} for 35 \degree \leq \alpha \leq 395 \degree.
Hence, solve \sin \left(\theta + 35 \degree\right) = \dfrac{1}{2}.
Consider the equation \sin \left(\theta + 25 \degree\right) = \dfrac{\sqrt{3}}{2} for 0 \degree \leq \theta \leq 360 \degree.
If \alpha = \theta + 25 \degree, find the solutions of \sin \alpha = \dfrac{\sqrt{3}}{2} for 25 \degree \leq \alpha \leq 385 \degree.
Hence, solve \sin \left(\theta + 25 \degree\right) = \dfrac{\sqrt{3}}{2}.
Consider the equation \tan \left(\theta + 15 \degree\right) = \dfrac{1}{\sqrt{3}} for 0 \degree \leq \theta \leq 360 \degree.
If \alpha = \theta + 15 \degree, find the solutions of \tan \alpha = \dfrac{1}{\sqrt{3}} for 15 \degree \leq \alpha \leq 375 \degree.
Hence, solve \tan \left(\theta + 15 \degree\right) = \dfrac{1}{\sqrt{3}}.
Solve the following equations for 0 \degree \leq \theta \leq 360 \degree:
\cos \left(\theta + 60 \degree\right) = 1
\cos \left(\theta - 75 \degree\right) = - \dfrac{1}{\sqrt{2}}
\sin \left(\theta + 45 \degree\right) = - \dfrac{\sqrt{3}}{2}
\tan \left(\theta - 45 \degree\right) = \dfrac{1}{\sqrt{3}}
\cos \left(\theta + 30 \degree\right) = -\dfrac{1}{2}
2\sin \left(\theta - 60 \degree\right) = -\sqrt{3}
Solve the following equations for 0 \degree \leq \theta \leq 360 \degree. Round your answers to two decimal places if necessary.
\cos 3 \theta = \dfrac{1}{\sqrt{2}}
\sin 2 \theta = -\dfrac{1}{2}
\cos 2 \theta = 0.7
2 \sin 3 \theta - \sqrt{2} = 0
\sin \left(\dfrac{\theta}{2}\right) = \dfrac{\sqrt{3}}{2}
\tan 3 \theta = - 1
Solve the following equations for 0 \degree \leq x \leq 180 \degree:
\tan 4 x = \sqrt{3}
\sin 2 x = \dfrac{\sqrt{3}}{2}
\cos 3 x = - \dfrac{1}{\sqrt{2}}
\tan\left(\dfrac{x}{3}\right) = 1
Solve the following equations for -180 \degree \leq x \leq 180 \degree:
\cos 3 x = - \dfrac{1}{\sqrt{2}}
\sin 2 x = \dfrac{1}{2}
\tan\left(\dfrac{x}{2}\right) = \pm \sqrt{3}
\sin 3x - 1 = 0
Solve the following equations for 0 \degree \leq \theta \leq 360 \degree. Round your answers to two decimal places if necessary.
2 \sin ^{2}\theta = 1
\text{cosec }^{2}\theta = 6.17
\cot ^{2}\theta = 0.24
\cos^{2}\left(\dfrac{\theta}{2}\right) - 1 = 0
\cos^{2} 3\theta = \dfrac{3}{4}
Solve the following equations for the given domain:
\cos ^{2}\theta = \dfrac{3}{4} for 0 \degree \lt \theta \lt 90 \degree
4 \sin ^{2}\theta = 3 for 90 \degree \leq \theta \leq 270 \degree
\sec ^{2}\theta = 2 for 450 \degree \leq \theta \leq 540 \degree
State the number of solutions for \theta of the equation \left(\sin \theta + \dfrac{\sqrt{3}}{2}\right) \left(\cos \theta + \dfrac{\sqrt{3}}{2}\right) = 0 for 0 \degree \lt \theta \lt 90 \degree.
Solve the following equations for 0 \degree \leq \theta \leq 360 \degree. Round your answer to one decimal place if necessary.
\left(\sec \theta + \sqrt{2}\right) \left(\text{cosec } \theta - 2\right) = 0
\left(\sec \theta + \dfrac{8}{5}\right) \left(\cot \theta + \dfrac{7}{4}\right) = 0
\left(\sec \theta + \sqrt{2}\right) \left(\text{cosec } \theta - 2\right) = 0
\left( \sqrt{3} \cot \theta - 1\right) \left(\cot \theta - 1\right) = 0
\tan ^{2}\left(\theta\right) - 2 \tan \theta + 1 = 0
\sec ^{2}\left(\theta\right) - \sec \theta - 42 = 0
\text{cosec } ^{2}\left(\theta\right) \cos \theta = 2 \cos \theta
4 \cos ^{3}\left(\theta\right) = 3 \cos \theta
5 \sin ^{2}\left(\theta\right) + 8 \sin \theta \cos \theta - 4 \cos ^{2}\left(\theta\right) = 0
\cot ^{2}\left(\theta\right) - 7 \cot \theta + 12 = 0
7 \cos ^{2}\left(\theta\right) + 3 \cos \theta = 3
9 \sin ^{2}\left(x\right) + 5 \sin x = 3
8 \cot ^{2}\left(\theta\right) - 5 \cot \theta = 1
\cos \theta \tan \theta = \cos \theta
8 \tan ^{2}\left(\theta\right) \cos \theta - 4 \tan ^{2}\left(\theta\right) = 0
Solve the following equations for 0 \degree \lt \theta \lt 90 \degree:
\left(\sin \theta - \dfrac{\sqrt{3}}{2}\right) \left(\cos \theta - \dfrac{1}{\sqrt{2}}\right) = 0
\left(\sin \theta + \dfrac{1}{\sqrt{2}}\right) \left(\tan \theta - \dfrac{1}{\sqrt{3}}\right) = 0
Solve the following equations for the given domain:
\left(\tan \theta - \dfrac{1}{\sqrt{3}}\right) \left(\cos \theta + \dfrac{1}{\sqrt{2}}\right) = 0 for 180 \degree \leq \theta \leq 270 \degree
\left(\sin \theta - \dfrac{1}{2}\right) \left(\cos \theta - \dfrac{1}{\sqrt{2}}\right) = 0 for 270 \degree \leq \theta \leq 360 \degree
\tan ^{2}\left(\theta\right) = \sqrt{3} \tan \theta for - 90 \degree \lt \theta \lt 90 \degree
2 \sin ^{2}\left(\theta\right) - \sin \theta = 0 for - 180 \degree \leq \theta \leq 180 \degree
Deborah is solving the equation 2 \sin^{2} \theta + 7 \sin \theta + 5 = 0. After some factorisation, she arrives at the pair of equations \sin \theta + 1 = 0 and 2 \sin \theta + 5 = 0. Which of the two equations has a solution?
Solve the equation 3 \sin \theta = 3 \sqrt{3} \cos \theta for 0 \lt \theta \lt 90.
Solve the following equations for 0 \degree \leq \theta < 360 \degree. Round your answers to the nearest degree if necessary.
2 \cos ^{2}\left(\theta\right) = 2 - \sin \theta
\sin ^{5}\left(\theta\right) \cos ^{3}\left(\theta\right) = 0
\cos ^{2}\left(\theta\right) - 8 \sin \theta \cos \theta + 3 = 0
\dfrac{1 - \tan ^{2}\left(\theta\right)}{1 + \tan ^{2}\left(\theta\right)} + \cos \theta = 0
2 \cos \theta - 1 = \sec \theta
\sin \theta - \text{cosec } \theta = 0
\tan ^{2}\left(\theta\right) + 5 = 3 \sec ^{2}\left(\theta\right)
6 \text{cosec }^{2}\left(\theta\right) = \cot \theta + 8
2 \sec ^{2}\left(\theta\right) - 3 \tan \theta = 11
Solve the following equations for the given domain:
3 \sqrt{3} \sec \theta = 3 \text{cosec } \theta for 270 \degree \leq \theta \leq 450 \degree
\tan ^{2}\left(x\right) + 9 = 3 \sec ^{2}\left(x\right) for 0 \degree \leq x \leq 360 \degree