27.05 Trigonometric equations (Extended)
Consider the function y = \cos \left(x - 30\right).
Sketch the graph of the function for 0 \leq x \leq 360.
Sketch the line y = \dfrac{1}{2} on the same number plane.
Hence, state all solutions to the equation \cos \left(x - 30\right) = \dfrac{1}{2} over the domain \left( 0 , 360\right].
Consider the function y = \sin \left(x - 60\right).
Sketch the graph of the function for 0 \leq x \leq 360.
Sketch the line y = \dfrac{1}{2} on the same number plane.
Hence, state all solutions to the equation \sin \left(x - 60\right) = \dfrac{1}{2} over the domain \left[0, 360\right).
Consider the function y = 2 \sin 4 x.
Sketch the graph of the function for 0\leq x \leq 120.
Sketch the line y = 1 on the same number plane.
Hence, state all solutions to the equation 2 \sin 4 x = 1 over the domain \left[ 0 , 90 \right]. Give your answers in degrees.
Consider the function y = 2 \sin 2 x.
Sketch the graph of the function for 0 \leq x \leq 180.
State the other function you would add to the graph in order to solve the equation 2 \sin 2 x = 1.
Sketch the graph of this function on the same number plane.
Hence, state all solutions to the equation 2 \sin 2 x = 1 over the domain \left[ - 180 \degree , 180 \degree\right].
Consider the function y = 3 \cos 2 x + 1.
Sketch the graph of the function for 0 \leq x \leq 180.
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[ 0 , 180\right].
Consider the function y = 2 \sin 3 x - 3.
Sketch the graph of the function for 0 \leq x \leq 60.
State the other function you would add to the graph in order to solve the equation 2 \sin 3 x - 3 = - 2.
Sketch the graph of this fuction on the same number plane.
Hence, state all solutions to the equation 2 \sin 3 x - 3 = - 2 over the domain \left[0 , 60\right].
Consider the function y = - 2 \cos 3 x.
Sketch the graph of the function for 0\degree \leq x \leq 120\degree.
State the other function you would add to the graph in order to solve the equation - 2 \cos 3 x = -1.
Sketch the graph of this function on the same number plane.
Hence, state all solutions to the equation - 2 \cos 3 x = -1 over the domain \left[ 0 \degree , 120 \degree\right].
State whether the following equations have a solution:
\cos \theta - 4 = 0
9 \tan \theta + 4 = 0
State the number of solutions for \theta of the following equations in the domain \\ 0 \degree \lt \theta \lt 90 \degree.
\cos \theta = - \dfrac{1}{\sqrt{2}}
\sin \theta = - \dfrac{\sqrt{3}}{2}
\tan \theta = - 1
Solve the following equations for 0 \degree \leq \theta \leq 90 \degree:
\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 \degree \leq \theta \leq 360 \degree:
\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
\cos \theta = 0.7986
\sin \theta =0.6428
\tan \theta =0.7265
\sin \theta = 0.3584
\tan \theta = 2.2460
Solve the following equations for 0 \degree \leq \theta \leq 360 \degree:
\cos \theta = 0.9063
\cos \theta = - 0.7986
\sin \theta = - 0.6428
\sin \theta = 0.9336
\tan \theta = 0.7002
\tan \theta = - 0.7265