Consider the functions y = 3 \sin x and y = 3.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2 \pi.
Hence, find the exact solutions to the equation 3 \sin x = 3 over the domain \left[ - 2 \pi , 2 \pi\right].
Consider the functions y = 3 \cos x and y = 3.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2 \pi.
Hence, find the exact solutions to the equation 3 \cos x = 3 over the domain \left[ - 2 \pi , 2 \pi\right].
Consider the functions y = \tan x and y = 1.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2 \pi.
Hence, find the exact solutions to the equation \tan x = 1 over the domain \left[ - 2 \pi , 2 \pi\right].
Consider the functions y = \sin \left(\dfrac{x}{4}\right) and y = - 0.5.
On the same set of axes, sketch the graphs of the functions for -4\pi \leq x \leq 4\pi.
Hence, find the exact solutions to the equation \sin \left(\dfrac{x}{4}\right) = - 0.5 over the domain \left[ - 4 \pi , 4 \pi\right].
Consider the functions y = \cos \left(\dfrac{x}{4}\right) and y = - 0.5.
On the same set of axes, sketch the graphs of the functions for -4\pi \leq x \leq 4\pi.
Hence, find the exact solutions to the equation \cos \left(\dfrac{x}{4}\right) = - 0.5 over the domain \left[ - 4 \pi , 4 \pi\right].
Consider the functions y = \tan \left(\dfrac{x}{2}\right) and y = 1.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation \tan \left(\dfrac{x}{2}\right) = 1 over the domain \left[ - 2 \pi , 2 \pi\right].
Consider the functions y = \sin \left(x - \dfrac{\pi}{3}\right) and y = \dfrac{1}{2}.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation \sin \left(x - \dfrac{\pi}{3}\right) = \dfrac{1}{2} over the domain \left[ - 2 \pi , 2 \pi\right).
Consider the functions y = \tan \left(x - \dfrac{\pi}{4}\right) and y = 1.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation \tan \left(x - \dfrac{\pi}{4}\right) = 1 over the domain \left[ - 2 \pi , 2 \pi\right).
Consider the functions y = \sin \left(x - \dfrac{\pi}{3}\right) + 5 and y = \dfrac{11}{2}.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation \sin \left(x - \dfrac{\pi}{3}\right) + 5 = \dfrac{11}{2} over the domain \left[ - 2 \pi , 2 \pi\right).
Consider the function y = - \sin \left(x + \dfrac{\pi}{4}\right) + 2 and y = 1.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation - \sin \left(x + \dfrac{\pi}{4}\right) + 2 = 1 over the domain \left( - 2 \pi , 2 \pi\right).
Consider the function y = - 2 \cos \left(x - \dfrac{\pi}{4}\right) - 2 and y = - 4.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation - 2 \cos \left(x - \dfrac{\pi}{4}\right) - 2 = - 4 over the domain \left( - 2 \pi , 2 \pi\right).
Consider the functions y = 2 \sin \left(x - \dfrac{\pi}{6}\right) + 5 and y = 6.
On the same set of axes, sketch the graphs of the functions for -2\pi \leq x \leq 2\pi.
Hence, find the exact solutions to the equation 2 \sin \left(x - \dfrac{\pi}{6}\right) + 5 = 6 over the domain \left[ - 2 \pi , 2 \pi\right).
Consider the function y = 3 \cos 2 x + 1.
State the equation of the second function you would graph in order to solve the equation 3 \cos 2 x + 1 = \dfrac{5}{2} graphically.
On the same set of axes sketch the graphs of the functions in the interval \left[ - \pi , \pi\right] .
Hence, find the exact solutions to the equation 3 \cos 2 x + 1 = \dfrac{5}{2} over the domain \left[ - \pi , \pi\right].
Consider the graph of y = \sin x in the interval \left[0, 2\pi\right]:
State the period of the graph.
Find the x-values for which \sin x = 0.
Find the lowest x-value for which \\ \sin x = 0.5.
Determine another value of x where \sin x = 0.5.
Find the values of x where \sin x = - 0.5.
Consider the graph of y = \cos x in the interval \left[0, 2\pi\right]:
State the period of the graph.
Find the x-values for which \cos x = 0.
Find the lowest x-value for which \\ \cos x = 0.5.
Determine another value of x where \cos x = 0.5.
Find the values of x where \cos x = - 0.5.
Consider the graph of y = \tan x in the interval \left[0, 2\pi\right]:
State the period of the graph.
Find the x-values for which \tan x = 0.
Find the lowest x-value for which \tan x = 1.
Determine another value of x where \tan x = 1.
Find the values of x where \tan x = - 1.
Consider the graph of y = \sin x + 3 in the interval \left[0, 2\pi\right]:
State the period of the graph.
Find the exact x-values for which \sin x + 3 = 3.
Find the lowest x-value for which \\ \sin x + 3 = 3.5.
Determine another value of x where \sin x + 3 = 3.5.
Find the values of x where \\ \sin x + 3 = 2.5.
Consider the graph of y = \cos x - 4 in the interval \left[-2\pi, 2\pi\right]:
State the period of the graph.
Find the x-values for which y = - 4.
Find the lowest positive x-value for which y = - 3.5.
Determine the other values of x where y = - 3.5.
Find the values of x where y = - 4.5.
State the exact value of x for which \cos x = \dfrac{\sqrt{3}}{2}.
Use the graph of the function \\y = \cos x and the answer to part (a) to find all other exact values of x between \\x = - \pi and x = \pi for which \\ \cos x = \pm \dfrac{\sqrt{3}}{2}.
Sketch the graphs of the functions y = - 3 \cos x and y = - 3 \sin x on the same coordinate axes, over the domain \left[ - 2 \pi , 2 \pi\right].
Hence, find the exact solutions to the equation - 3 \cos x = - 3 \sin x over the domain \left[ - 2 \pi , 2 \pi\right].
Use the equation - 3 \sin x = - 3 \cos x to show that \tan x = 1.
State the exact solutions to \tan x = 1 over the domain \left[ - 2 \pi , 2 \pi\right].
Consider the equation 3 \sin \left( 3 x + \dfrac{\pi}{7}\right) = - \dfrac{11}{10}.
Determine the function that must be graphed along with y = - \dfrac{11}{10} in order to solve the equation graphically.
Use technology to graph both of these functions. Hence, find all solutions to the equation over the domain \left[ - \dfrac{13}{42} \pi, \dfrac{5}{14} \pi\right], correct to three decimal places.
Consider the equation - 5 \cos \left(\dfrac{x}{2} + \dfrac{\pi}{5}\right) = - \dfrac{17}{10}.
Determine the function that must be graphed along with y = - \dfrac{17}{10} in order to solve the equation graphically.
Use technology to graph both of these functions. Hence, find all solutions to the equation over the domain \left[ - \dfrac{7 \pi}{5} , \dfrac{13 \pi}{5}\right], correct to three decimal places.
Find the measure in radians of the acute angle satisfying the following equations, correct to two decimal places.
\sin \theta = 0.3675
\cos \theta = 0.7482
\tan \theta = 2.5869
For each of the following equations:
Find the measure in radians of the acute angle satisfying the equation, correct to two decimal places.
Find the measure in radians of the angles satisfying the equation in the interval \\2 \pi \leq \theta \leq 4 \pi, correct to two decimal places.
\sin \theta = 0.2756
\cos \theta = 0.3746
Find the measure in radians of the acute angles satisfying the following equations, correct to two decimal places:
2 \sin \theta + 3 = 6 \sin \theta
4 \cos \theta + 4 = 9 \cos \theta
5 \tan \theta + 4 = 9 \tan \theta - 1
3 \left(\sin \theta + 1\right) = 5 - 11 \sin \theta
\sin ^{2}\left(\theta\right) = 0.46
\tan ^{2}\left(\theta\right) = 3.62
6 \sin \theta = 12.84 \cos \theta
Find the measure in radians of the angle(s) satisfying the following equations for 0 < \theta < \dfrac{\pi}{2}, correct to two decimal places:
\left(\sin \theta - \dfrac{5}{8}\right) \left(\cos \theta - \dfrac{2}{3}\right) = 0
12 \sin ^{2}\left(\theta\right) - 11 \sin \theta + 2 = 0
The point P\left(x, y\right) corresponds to a point on the unit circle at a rotation of \theta, such that \sin \theta = \dfrac{3}{5} and \cos \theta = - \dfrac{4}{5}. In which quadrant is P located?
Suppose an angle of rotation \alpha = 3.5^{c} corresponds to the point P\left(x, y\right) on the unit circle:
In which quadrant is \left(x, y\right) located?
State the sign of \sin \alpha.
State the sign of \cos \alpha.
State the sign of \tan \alpha.
How many solutions for \theta do the following equations have for 0 < \theta < \dfrac{\pi}{2}?
\sin \theta = 0.3374
\tan \theta = - 0.5561
\cos \theta = \dfrac{1}{3}
\sin \theta = -\dfrac{1}{2}
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 ^2 {\theta} + 1= 0
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 exact measure of the angles satisfying the following equations for 0 \leq \theta \leq \dfrac{\pi}{2}:
\cos \theta = \dfrac{\sqrt{3}}{2}
10 \sin \theta - 5 = 0
\tan \theta = \dfrac{2}{\sqrt{3}} - \tan \theta
Find the exact measure of the angles satisfying the following equations for 0 \leq \theta \leq 2 \pi:
\cos \theta = - \dfrac{1}{2}
\tan \theta = - \dfrac{1}{\sqrt{3}}
10 \sin \theta - 5 \sqrt{3} = 0
For the following equations, find the exact values of x in the interval \left[0, 2 \pi\right):
6 \cos x - 3 \sqrt{2} = 0
2 \sin ^{2}\left(x\right) = 1
\sin ^{2}\left(x\right) - 6 \cos ^{2}\left(x\right) = 1
4 \sin x \cos x = \sqrt{3}
For the following equations, find the exact values of x in the domain 0 \leq x \leq 2 \pi:
\sin 2 x = \dfrac{1}{\sqrt{2}}
\tan 3 x = - \dfrac{1}{\sqrt{3}}
\sqrt{3} \tan \left(\dfrac{x}{2}\right) = - 3
For the following equations, find the exact values of x in the domain of 0 \leq x \leq \pi:
\cos 4 x = - \dfrac{\sqrt{3}}{2}
\sin 5 x = - \dfrac{1}{2}
\tan 4 x = \dfrac{1}{\sqrt{3}}
Find the exact solutions to the equation \tan 4 x = \sqrt{3} in the domain -\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}.
Find the exact solutions to the equation \cos 3 x = - \dfrac{\sqrt{3}}{2} in the domain -\pi \leq x \leq \pi.
Find the exact solutions to the equation \sin \left(\dfrac{x}{2}\right) = \dfrac{\sqrt{3}}{2} in the interval [0,\pi).
For the following equations, find the exact values of x in the interval [0,2\pi):
\cos \left(\dfrac{x}{2}\right) = 1 - \cos \left(\dfrac{x}{2}\right)
\cos ^{2}\left(\dfrac{x}{2}\right) - 1 = 0
Find the exact solutions for each equation in the given interval:
\cos \left(x - \dfrac{\pi}{5}\right) = \dfrac{\sqrt{3}}{2} for 0 \leq x \leq 3 \pi.
2 \sin \left(\dfrac{x}{2} + \dfrac{\pi}{5}\right) = 1 for - 4 \pi \leq x \leq 4 \pi.
\sin \left( 3 x - \dfrac{\pi}{3}\right) = - 1 for - \pi \leq x \leq \pi.
\sqrt{2} \cos \left(x - \dfrac{\pi}{3}\right) = 1 for - 2 \pi \leq x \leq 2 \pi.
\tan \left(x + \dfrac{\pi}{5}\right) = \sqrt{3} for 0 \leq x \leq 2 \pi.
\tan \left( \pi x - \dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{3}} for - 2 \leq x \leq 2.
2 \tan \left( 4 x + \dfrac{\pi}{5}\right) = 1 for - \dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}.