5.02 Unit circle
Calculate the fraction of the circumference of the unit circle the following angle measures represent:
x = 60 \degree
x = 180 \degree
s = 450 \degree
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
Write four different angles between 0 \degree and 360 \degree inclusive, that lie on the quadrant boundaries.
For each of the following circular functions:
State the quadrant(s) in which the function is positive.
State the quadrant(s) in which the function is negative.
Sine
Cosine
Tangent
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.
State whether the values of the following are positive or negative:
\sin 31 \degree
\tan 31 \degree
\cos 267 \degree
\sin 267 \degree
\cos 180 \degree
\tan 296 \degree
\sin 120 \degree
\cos 91 \degree
\sin 296 \degree
\cos 120 \degree
\cos 296 \degree
\sin 90 \degree
\cos 51 \degree
\sin 51 \degree
\cos 233 \degree
\tan 233 \degree
For each of the following graphs, state:
\sin a
\cos a
\tan a
Determine whether the following statements are possible:
\sin \theta = \dfrac{7}{8}
\sin \theta = - 0.6
\sin \theta = 1.1
\sin \theta = - \dfrac{2}{3}
Consider the following angles:
w = 253 \degree, \quad x = 265 \degree, \quad y = 193 \degree, \quad z = -258 \degree, \quad a = -182 \degree, \quad b = -257 \degree
Determine whether the following statements are true or false:
Angles 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 all have negative cosine values.
Angles w, x, and y are in Quadrant 3.
Consider the following angles:
w = 65 \degree, \quad x = 10 \degree, \quad y = 44 \degree, \quad z = -31 \degree , \quad a = -10 \degree , \quad b = -29 \degree
Determine whether the following statements are true or false:
\tan z, \cos \left(90 \degree + x\right) and \sin \left(180 \degree + y\right) are negative.
\sin x, \cos z, and \tan b are all positive.
Angles w, x, and y are in Quadrant 1.
\sin x, \cos y, and \tan w are all positive.
Angles z, a, and b are in Quadrant 1.
The terminal side of an angle, \theta passes through the point \left( - 4 , - 6 \right) as shown in the diagram:
Find the value of r, the distance between \left(0, 0\right) and \left( - 4 , - 6 \right).
Hence find the exact values of the following:
\sin \theta
\cos \theta
\tan \theta
Suppose that \cos \theta = \dfrac{3}{5}, where \\ 270 \degree < \theta < 360 \degree. Find the exact value of the following:
\sin \theta
\tan \theta
Suppose that \sin \theta = - \dfrac{\sqrt{7}}{4}. Find the exact value of the following:
\cos \theta
\tan \theta
Given angle \theta such that \sin \theta = 0.6 and \tan \theta < 0, find the exact value of the following:
\cos \theta
\tan \theta
Given angle \theta such that \tan \theta = - 1.3 and 270 \degree < \theta < 360 \degree, find the exact value of the following:
\cos \theta
\sin \theta
Given that \tan \theta = - \dfrac{15}{8} and \sin \theta > 0, find the exact value of \cos \theta.
Given the following, find the exact value of \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 exact value 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
Consider the ratio \sin 150 \degree.
State the quadrant in which 150 \degree is located.
State whether \sin 150 \degree is positive or negative.
Find the positive acute angle that 150 \degree is related to.
Hence rewrite \sin 150 \degree in terms of a trigonometric ratio of a positive acute angle.
Consider the ratio \cos 150 \degree.
State the quadrant in which 150 \degree is located.
State whether \cos 150 \degree is positive or negative.
Find the positive acute angle that 150 \degree is related to.
Hence rewrite \cos 150 \degree in terms of a trigonometric ratio of a positive acute angle.
Consider the ratio \tan 330 \degree.
State the quadrant in which 330 \degree is located.
State whether \tan 330 \degree is positive or negative.
Find the positive acute angle 330 \degree is related to.
Hence rewrite \tan 330 \degree in terms of a trigonometric ratio of a positive acute angle.
For each of the following, rewrite the ratio as an equivalent trigonometric ratio of a positive acute angle:
\sin 93 \degree
\cos 195 \degree
\tan 299 \degree
\sin 240 \degree
\sin 147 \degree
\tan \left( - 50 \degree \right)
\sin 400 \degree
\cos 605 \degree
\tan 525 \degree
\sin \left( - 139 \degree \right)
\cos \left( - 222 \degree \right)
\tan \left( - 289 \degree \right)
\sin \left( - 221 \degree \right)
Consider the following diagram of a unit circle with centre O.
Express the following trigonometric ratios in terms of a and/or b:
\tan 65 \degree
\cos 475 \degree
\sin 65 \degree
\cos 65 \degree
\cos 295 \degree
\sin 245 \degree
The diagram shows points P, Q, R and S, which represent rotations of 49 \degree, 131 \degree, 229 \degree and 311 \degree respectively around the unit circle.
If point P has coordinates \left(a, b\right), write the coordinates of the following points in terms of a and b:
Q
R
S
State whether the following ratios are equivalent to \sin 49 \degree or - \sin 49 \degree:
\sin 131 \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, Q, R and S, which represent rotations of 63 \degree, 117 \degree, 243 \degree and 297 \degree respectively around the unit circle.
If point P has coordinates \left(a, b\right), write the coordinates of the following points in terms of a and b:
Q
R
S
State whether the following are equivalent to - \cos 63 \degree or \cos 63 \degree:
\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, Q, R and S, which represent rotations of 62 \degree, 118 \degree, 242 \degree and 298 \degree respectively around the unit circle.
If point P has coordinates \left(a, b\right), write the coordinates of the following points in terms of a and b:
Q
R
S
State whether the following are equivalent to - \tan 62 \degree or \tan 62 \degree:
\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)
Simplify:
\sin \left( - \theta \right)
\cos \left( - \theta \right)
Given the approximations \cos 21 \degree = 0.93 and \sin 21 \degree = 0.36, state the approximate values of the following, to 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
Find the following, correct to two decimal places:
\sin 146 \degree
\cos 126 \degree
\tan 161 \degree
\tan 386 \degree
\cos 621 \degree
\cos (-218\degree)
\sin (-42.5\degree)
-\tan (-89\degree)
\left(\sin 35 \degree \right)^{2}
\sin ^{2} 35 \degree
\sin \left( 35 \degree \right)^{2}
-\sin ^{2} 35 \degree
Find the following:
\sin ^{2} 35 \degree + \cos ^{2} 35 \degree
\sin ^{2} \left(-64 \right) \degree + \cos ^{2} \left(-64 \right) \degree
\sin ^{2} 275 \degree + \cos ^{2} 275 \degree
\sin ^{2} x \degree + \cos ^{2} x \degree
Consider the following diagram of a unit circle:
Given that point B, represents a rotation of 35 \degree around the unit circle, find the coordinates of B, correct to three decimal places.
Point C has coordinates \left(0.294, 0.956\right). Find the angle that point C represents to the nearest degree.
Given that the angle subtended by arc \angle CD = 54 \degree, find the coordinates of D, correct to three decimal places.
The diagram shows P, which represents a rotation of 66 \degree around the unit circle. Find the following, rounding your answers to two decimal places:
Coordinates of point P.
Coordinates of point R, that represents point P reflected horizontally about the y-axis.
Size of the rotational angle for R around the unit circle.
Coordinates of point Q, that represents point P reflected vertically about the \\ x-axis.
Size of the rotational angle for Q around the unit circle.
