Solve the following equations for x:
Solve the equation \sin ^{-1}x - \tan ^{-1}1 = - \dfrac{\pi}{4} for x.
Consider the equation \cos ^{-1}x + 2 \sin ^{-1}\left(\dfrac{1}{\sqrt{2}}\right) = \dfrac{\pi}{4}.
Solve for \cos ^{-1}x.
Does the equation have a solution? Explain your answer.
Consider the equation 4 \tan ^{-1}x = - 3 \pi.
Solve the equation for \tan ^{-1}x.
Does the equation have a solution? Explain your answer.
Solve for the exact value of x in the equation \sin ^{-1}x = \tan ^{-1}\dfrac{15}{8}.
Solve y = 2 \cos x for x in terms of y, where x is in \left[0,\pi\right].
Solve y = 5 \tan 3 x for x in terms of y, where x is in \left( - \dfrac{\pi}{6}, \dfrac{\pi}{6} \right).
Solve y = - 5 \cos 3 x for x in terms of y, where x is in \left[0,\dfrac{\pi}{3}\right].
Solve y = - 2 + 3 \sin x for x in terms of y, where x is in \left[ - \dfrac{\pi}{2} , \dfrac{\pi}{2}\right].
Solve y = \tan \left( 3 x - 1 \right) for x in terms of y, where x is in \left(\dfrac{1}{3} - \dfrac{\pi}{6}, \dfrac{1}{3} + \dfrac{\pi}{6} \right).
Consider the equation \cos \theta = 0.42. State the number of solutions for \theta over the domain \\ 0 \lt \theta \lt 2\pi.
Consider the equation \sin \theta = - 0.351 for all real values of \theta. How many solutions for \theta does the equation have?
State whether it is possible to find values of \theta that satisfy the following equations:
\cos \theta - 4 = 0
4 \cos \theta - 3 = 0
9 \tan \theta + 4 = 0
\sin \theta = 5
\sin \theta = \sqrt{2}
\tan \theta = - 99.92
Solve the following equations for 0 \degree \leq \theta \leq \dfrac{\pi}{2}:
\sin \theta = \dfrac{1}{\sqrt{2}}
\tan \theta = \sqrt{3}
\cos \theta = \dfrac{1}{2}
\sin \theta = \dfrac{\sqrt{3}}{2}
For each of the following, find the exact values of \theta, in the given domain:
\tan \theta = 0, where - \pi \leq \theta \leq \pi
\tan \left(x - \dfrac{2 \pi}{3}\right) = 0, where - \pi < x < \pi
For each of the following, find the exact values of x, over the domain 0 \leq x \leq 2 \pi:
\sin x = 1
\cos x = - 1
\sin x = \dfrac{1}{2}
\cos x = - \dfrac{\sqrt{3}}{2}
\cos x = - \dfrac{1}{\sqrt{2}}
\cos x = \dfrac{1}{2}
\cos x = 0
\sin x = 0
\sin x = - \dfrac{1}{\sqrt{2}}
\sin x = - \dfrac{\sqrt{3}}{2}
\tan x = 0
\tan x = \sqrt{3}
\tan x = - \dfrac{1}{\sqrt{3}}
4 \tan x + 2 = - 2
8 \cos x - 4 = 0
2 \cos x + 4 = 3
8 \sin x - 4 \sqrt{2} = 0
6 \cos x - 3 \sqrt{2} = 0
2 \tan x + 3 = 5
2 \sin x + 2 = 1
6 \cos x + 2 = - 1
10 \sin x - 5 \sqrt{3} = 0
8 \sin x + 4 = 0
\sin x = - \cos x
For each of the following, find the exact values of x, over the domain \left[ 0 , 2 \pi \right):
\sin 2 x = \dfrac{1}{\sqrt{2}}
2 \sin 3 x - \sqrt{2} = 0
\sin \left(\dfrac{x}{2}\right) = \dfrac{\sqrt{3}}{2}
2 \sin \left(\dfrac{x}{2}\right) = \sqrt{3}
\sin \left(\dfrac{x}{2}\right) = - \cos \left(\dfrac{x}{2}\right)
\cos \left(\dfrac{x}{2}\right) = 1 - \cos \left(\dfrac{x}{2}\right)
\sin \left(\dfrac{x}{2}\right) = 1 - \sin \left(\dfrac{x}{2}\right)
\sqrt{3} \tan \left(\dfrac{x}{2}\right) = - 3
2 \sin ^{2}\left(x\right) = 1
\cos ^{2}\left(\dfrac{x}{2}\right) - 1 = 0
\sin ^{2}\left(\dfrac{x}{2}\right) - 1 = 0
For each of the following, find the exact values of x, over the given domain:
\cos \left(x + \dfrac{\pi}{6}\right) = - \dfrac{1}{2}
\sin \left(x + \dfrac{\pi}{6}\right) = \dfrac{1}{\sqrt{2}}
\sin \left(x - \dfrac{4 \pi}{3}\right) = \dfrac{1}{2}
\sqrt{2} \cos \left(x - \dfrac{\pi}{3}\right) = 1 for - 2 \pi \leq x \leq 2 \pi
Solve \cos 2 x = 0.9 for 0 \leq x < 2 \pi. Give your answers correct to two decimal places.
Find the angle satisfying \cos \theta = 0.7482 for 0 < \theta < \dfrac{\pi}{2}. Round your answer correct to two decimal places.
Find the acute angle satisfying 2 \sin \theta + 3 = 6 \sin \theta. Round your answer to two decimal places.
Find the acute angle satisfying 5 \tan \theta + 4 = 9 \tan \theta - 1. Round your answer to three decimal places.
Find the angle satisfying \sin ^{2}\left(\theta\right) = 0.46 for 0 < \theta < \dfrac{\pi}{2}. Round your answer to two decimal places.
Consider the equation \sin \theta = 0.2756.
Find the acute angle satisfying the equation. Round your answer to two decimal places.
Find the angles satisfying \sin \theta = 0.2756 for 2 \pi \leq \theta \leq 4 \pi. Round your answers to two decimal places.
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 these two equations has a solution?
Hence, determine the exact solutions to the equation, for 0 \leq \theta \leq 2\pi.
Neville is solving the equation \cos^{2} \theta - 5 \cos \theta + 4 = 0. After some factorisation, he arrives at the pair of equations \cos \theta - 4 = 0 and \cos \theta - 1 = 0.
Which of these two equations has a solution?
Hence, determine the exact solutions to the equation, for 0 \leq \theta \leq 2\pi.
Find the angles satisfying 12 \sin ^{2}\left(\theta\right) - 11 \sin \theta + 2 = 0 for 0 < \theta < \dfrac{\pi}{2}. Round your answers to three decimal places.
For each of the following, find the exact values of x, over the domain 0 \leq x \leq 2 \pi:
\tan ^{2}\left(x\right) + 2 \tan x + 1 = 0
\cos x \tan x = \cos x