Consider the function y = 3 \sin x and the line y = 3 which are graphed below:
State all solutions to the equation 3 \sin x = 3 over the domain \left[ - 2 \pi , 2 \pi\right].
Graph the function y = \tan x and the line y = 1 on the same number plane over the domain \left[ - 2 \pi , 2 \pi\right].
Hence, state all solutions to the equation \tan x = 1 over the domain \left[ - 2 \pi , 2 \pi\right].
Consider the graph of y = \cos x over the domain [0, 2 \pi ]:
State the x-values for which \cos x = 0.
State the first x-value for which \cos x = 0.5
For what other value of x shown on the graph, does \cos x = 0.5?
For what values of x does \cos x = - 0.5?
Consider the graph of y = \tan x:
How long is one period of the graph?
State the x-values for which \tan x = 0, from x = 0 to x = 2 \pi inclusive.
State the first x-value for which \tan x = 1.
For what other value of x shown on the graph does \tan x = 1?
For what values of x shown on the graph does \tan x = - 1?
Consider the function y = \cos \left(\dfrac{x}{4}\right).
Sketch the graph of this function over the domain [-4 \pi, 4\pi].
Sketch the line y = - 0.5 on the same number plane.
Hence, state all solutions to the equation \cos \left(\dfrac{x}{4}\right) = - 0.5 over the domain \left[ - 4 \pi , 4 \pi\right] in exact form.
Consider the function y = \tan \left(x - \dfrac{\pi}{4}\right).
Sketch the graph of the function for -2\pi \leq x \leq 2\pi.
Sketch the line y = 1 on the same number plane.
Hence, state all solutions to the equation \tan \left(x - \dfrac{\pi}{4}\right) = 1 over the domain \left[ - 2 \pi , 2 \pi\right). Give your answers as exact values.
Consider the function y = 3 \cos 2 x + 1.
Sketch the graph of the function for -\pi \leq x \leq \pi.
State the other function you would add to the graph in order to solve the equation 3 \cos 2 x + 1 = \dfrac{5}{2}.
Sketch the graph of this function on the same number plane.
Hence, state all solutions to the equation 3 \cos 2 x + 1 = \dfrac{5}{2} over the domain \left[ - \pi , \pi\right]. Give your answers as exact values.
Consider the function y = \sin \left(x - \dfrac{\pi}{3}\right) + 5.
Sketch the function y = \sin \left(x - \dfrac{\pi}{3}\right) + 5 over the domain [- 2 \pi, 2 \pi].
Sketch the line y = \dfrac{11}{2} on your graph.
Hence, state all solutions to the equation \sin \left(x - \dfrac{\pi}{3}\right) + 5 = \dfrac{11}{2} over the domain \left[ - 2 \pi , 2 \pi\right). Give your answers in exact form.
Consider the equation 3 \sin \left( 3 x + \dfrac{\pi}{7}\right) = - \dfrac{11}{10}.
Which function would be graphed along with y = - \dfrac{11}{10} in order to solve the equation graphically?
Graph both of these functions using the graphing facility of your calculator. Hence state all solutions to the equation over the domain \left[ - \dfrac{13\pi}{42}, \dfrac{5\pi}{14}\right]. Round your answers correct to three decimal places.
Consider the equation - 5 \cos \left(\dfrac{x}{2} + \dfrac{\pi}{5}\right) = - \dfrac{17}{10}.
Which function would be graphed along with y = - \dfrac{17}{10} in order to solve the equation graphically?
Graph both of these functions using the graphing facility of your calculator. Hence state all solutions to the equation over the domain \left[ - \dfrac{7 \pi}{5} , \dfrac{13 \pi}{5}\right]. Round your answers correct to three decimal places.
Explain whether the following equations have a solution:
\cos \theta - 4 = 0
2\tan \theta + 4 = 0
Solve the following equations for 0 \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}
Solve the following equations for 0 \leq \theta \leq 2\pi :
\cos \theta = - \dfrac{1}{\sqrt{2}}
\cos \theta = \dfrac{1}{2}
\cos \theta = 0
\sin \theta = \dfrac{1}{2}
\sin \theta = 0
\sin \theta = - \dfrac{1}{\sqrt{2}}
\cos \theta = -\dfrac{1}{\sqrt{2}}
\sin \theta = - \dfrac{\sqrt{3}}{2}
\sin \theta = 1
\tan \theta = \sqrt{3}
\tan \theta = 0
\tan \theta = - \dfrac{1}{\sqrt{3}}
4 \tan \theta + 2 = - 2
8 \cos \theta - 4 = 0
2 \cos \theta + 4 = 3
8 \sin \theta - 4 \sqrt{2} = 0
Solve for x in the domain of 0 \leq x \leq 2 \pi:
\sin 2 x = \dfrac{1}{\sqrt{2}}
10 \sin x - 5 \sqrt{3} = 0
\cos \left(\dfrac{x}{2}\right) = 1 - \cos \left(\dfrac{x}{2}\right)
Solve each equation for the given interval. Give your answers in exact form.
\sqrt{2} \cos \left(x - \dfrac{\pi}{3}\right) = 1 for - 2 \pi \leq x \leq 2 \pi
Solve \sin ^{2}\left(x\right) - 6 \cos ^{2}\left(x\right) = 1 over the interval [0,2 \pi).
Find the angle satisfying \cos \theta = 0.7482 for 0 < \theta < \dfrac{\pi}{2}. Round your answer correct 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.
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.
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?
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.