5.03 The unit circle and angles of any magnitude
State the quadrant in which the following angles are located:
299\degree
5\degree
160\degree
229\degree
40\degree
310\degree
- 138\degree
- 244\degree
For the given functions, state the following:
The quadrants where the function is positive.
The quadrants where the function is negative.
Sine function
Cosine function
Tangent function
State whether the values of the following are positive or negative:
\sin 310 \degree
\sin 50 \degree
\sin 130 \degree
\sin 230 \degree
\sin 31 \degree
\tan 31 \degree
\cos 267 \degree
\sin 267 \degree
\cos 180 \degree
Consider the given angles below and state whether the following are true or false:
w=253\degree,\quad x=265\degree,\quad y=193\degree, \quad z=-258\degree,\quad a=-182\degree,\quad b=-257\degreeAngles z, a, and b are in Quadrant 3.
\sin z and \sin b are both negative.
\tan \left(z - 90 \degree\right) and \tan \left(z + 90 \degree\right) are both positive.
The angles w, x, and y have negative cosine values.
Angles w, x, and y are in Quadrant 3.
State the quadrant where the point in each scenario is located:
The point \left(x, y\right) is on the unit circle at a rotation of \theta such that \sin \theta = \dfrac{5}{13} and \cos \theta = \dfrac{12}{13}.
The point P\left(x, y\right) is on the unit circle at a rotation of \theta such that \sin \theta = - \dfrac{12}{37} and \cos \theta = \dfrac{35}{37}.
State the quadrant where the angle in each scenario is located:
\theta is an angle such that \sin \theta > 0 and \cos \theta < 0.
\theta is an angle such that \tan \theta < 0 and \sin \theta > 0.
\theta is an angle such that \tan \theta < 0 and \cos \theta < 0.
\theta is an angle such that \tan \theta > 0 and \sin \theta > 0.
Point P on the unit circle shows a rotation of 330 \degree. Find the related acute angle in the first quadrant.
The diagram shows points P \left(a, b\right), Q, R and S, which represent rotations of 49 \degree, 131 \degree, 229 \degree and 311 \degree respectively around the unit circle.
Find the coordinates of the following points in terms of a and b:
Q
R
S
Find the equivalent trigonometric ratio in the first quadrant of the following:
\sin 131 \degree
\sin 229 \degree
\sin 311 \degree
Hence, express each of the following in terms of \sin x:
\sin \left(180 \degree - x\right)
\sin \left(180 \degree + x\right)
\sin \left(360 \degree - x\right)
The diagram shows points P \left(a, b\right), Q, R and S, which represent rotations of 63 \degree, 117 \degree, 243 \degree and 297 \degree respectively around the unit circle.
Find the coordinates of the following points in terms of a and b:
Q
R
S
Find the equivalent trigonometric ratio in the first quadrant of the following:
\cos 117 \degree
\cos 243 \degree
\cos 297 \degree
Hence, express each of the following in terms of \cos x:
\cos \left(180 \degree - x\right)
\cos \left(180 \degree + x\right)
\cos \left(360 \degree - x\right)
The diagram shows points P \left(a, b\right), Q, R and S, which represent rotations of 62 \degree, 118 \degree, 242 \degree and 298 \degree respectively around the unit circle.
Find the coordinates of the following points in terms of a and b:
Q
R
S
Find the equivalent trigonometric ratio in the first quadrant of the following:
\tan 118 \degree
\tan 242 \degree
\tan 298 \degree
Hence, express each of the following in terms of \tan x:
\tan \left(180 \degree - x\right)
\tan \left(180 \degree + x\right)
\tan \left(360 \degree - x\right)
For each of the following graphs, find:
\sin a
\cos a
\tan a
The following are angle rotations at point P on the unit circle. Find the acute angle in the first quadrant related to each rotation:
295 \degree
- 150 \degree
240 \degree
- 75 \degree
Rewrite each expression as a trigonometric ratio of a positive acute angle:
\sin 93 \degree
\cos 195 \degree
\tan 299 \degree
\sin \left( - 139 \degree \right)
\cos \left( - 222 \degree \right)
\tan \left( - 289 \degree \right)
Simplify:
\sin \left( - \theta \right)
\cos \left( - \theta \right)
Given the approximations \cos 21 \degree = 0.93 and \sin 21 \degree = 0.36, find the approximate values of the following and giving your answers in two decimal places:
\cos 339 \degree
\sin \left( - 339 \degree \right)
\cos 159 \degree
\sin 159 \degree
\sin 201 \degree
\cos 201 \degree
\sin (-201) \degree
\cos (-201) \degree
Evaluate the following correct to two decimal places:
\sin 146 \degree
\tan 386 \degree
\cos 387 \degree
The diagram shows P, which represents a rotation of 66 \degree around the unit circle:
Find the following to two decimal places:
Coordinates of P(x,y)
The new coordinates of P if it was reflected about the y-axis.
\sin 114 \degree
\cos 114 \degree
\tan 114 \degree
Suppose that \cos \theta = \dfrac{3}{5}, where \\ 270 \degree < \theta < 360 \degree.
Find \sin \theta.
Find \tan \theta.
Suppose that \sin \theta = - \dfrac{\sqrt{7}}{4}.
Find \cos \theta.
Find \tan \theta.
Consider an angle \theta such that \sin \theta = 0.6 and \tan \theta < 0, find:
\sin \theta
\cos \theta
\tan \theta
Consider an angle \theta such that \tan \theta = - 1.3 and 270 \degree < \theta < 360 \degree, find:
\cos \theta
\sin \theta
Given that \tan \theta = - \dfrac{15}{8} and \sin \theta > 0, find \cos \theta.
Given the following, find \sin \theta.
\cos \theta = - \dfrac{60}{61} and 0 \degree \leq \theta \leq 180 \degree
\cos \theta = - \dfrac{6}{7} and \tan \theta < 0
Given the following, find the value(s) of \tan \theta.
\cos \theta = \dfrac{3}{7} and \theta is acute
\sin \theta = \dfrac{1}{\sqrt{10}} and - 90 \degree \leq \theta \leq 90 \degree
The first diagram shows a unit circle with point P \left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) marked on the circle. Point P represents a rotation of 60 \degree anticlockwise around the origin from the positive x-axis:
Find the exact values of the following:
\sin 60\degree
\cos 60\degree
\tan 60\degree
On the second diagram, the coordinate axes shows a 60 \degree angle that has also been marked in the second, third, and fourth quadrants. For each quadrant, find the relative angle.
Quadrant 2
Quadrant 3
Quadrant 4
The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. Find the exact coordinates of each point:
Q
R
S
The first diagram shows a unit circle with point P \left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) marked on the circle. Point P represents a rotation of 30 \degree anticlockwise around the origin from the positive x-axis:
Find the exact values of the following:
\sin 30\degree
\cos 30\degree
\tan 30\degree
On the second diagram, the coordinate axes shows a 30 \degree angle that has also been marked in the second, third and fourth quadrants. For each quadrant, find the relative angle:
Quadrant 2
Quadrant 3
Quadrant 4
The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. Find the exact coordinates of each point:
Q
R
S
Consider the trigonometric ratio \cos \left( - 210 \degree \right).
Find the value of the related acute angle.
Hence, find the value of \cos \left( - 210 \degree \right) using exact values.
Find the exact value of the following:
\sin 225 \degree
\cos 225 \degree
\tan 225 \degree
\sin 690 \degree
\sin 135 \degree
\cos 315 \degree
\tan 150 \degree
\sin 240 \degree
\cos 690 \degree
\tan 690 \degree
\sin 50 \degree + \sin 160 \degree + \sin 200 \degree + \sin 310 \degree
\tan 60 \degree + \tan 315 \degree
\tan ^{2}\left(240 \degree\right) - \sin ^{2}\left(210 \degree\right) + \cos ^{2}\left(180 \degree\right)
By rewriting each ratio in terms of the related acute angle, find the exact value of:
\dfrac{\sin 120 \degree \cos 240 \degree \tan 330 \degree}{\tan \left( - 45 \right) \degree}