6.01 Exact ratios and the unit circle
Find the exact value of the following:
\sin 30 \degree
\sin 60 \degree
\sin 45 \degree
\cos 45 \degree
\cos 60 \degree
\cos 30 \degree
\tan 60 \degree
\tan 45 \degree
\tan 30 \degree
Evaluate the following expressions, leaving your answer in exact, rationalised form:
Find the exact value of the pronumeral in the following triangles:
Consider the given triangle:
Find the exact length of side a.
Find the exact length of side b.
For each of the following triangles:
Find the exact value of x.
Find the exact value of y.
Consider the unknown lengths in the given figure:
Find m.
Find t.
\theta is an angle in a right-angled triangle. Find the value of \theta in the following equations:
\cos \theta = \dfrac{1}{2}
\sin \theta = \dfrac{\sqrt{3}}{2}
\sin \theta = \dfrac{1}{\sqrt{2}}
\tan \theta = \dfrac{1}{\sqrt{3}}
Find the value of \theta if \cos \theta = \dfrac{1}{\sqrt{2}} and \sin \theta = \dfrac{1}{\sqrt{2}}.
Given that \cos \theta = \dfrac{\sqrt{3}}{2} and \sin \theta = \dfrac{1}{2}:
Find the value of \theta.
Find the value of \tan \theta.
Find the unknown \theta in the following triangles:
State the quadrant in which the following angles are located:
299 \degree
5 \degree
160\degree
229\degree
40\degree
310\degree
138\degree
344\degree
Write four different angles between 0 \degree and 360 \degree inclusive, that lie on the quadrant boundaries.
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
Consider the point \left(x, y\right) on the following unit circles:
Find the value of x.
Find the value of y.
Consider the point on the given unit circle. Find the value of \theta.
The first diagram shows a unit circle with point P \left(\dfrac{1}{\sqrt2}, \dfrac{1}{\sqrt2}\right) marked on the circle. Point P represents a rotation of 45 \degree anticlockwise around the origin from the positive x-axis:
Find the exact values of the following:
\sin 45\degree
\cos 45\degree
\tan 45\degree
On the second diagram, the coordinate axes shows a 45 \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. State the exact coordinates of each point:
Q
R
S
Write the following in terms of an equivalent ratio of 45 \degree:
\sin 135\degree
\cos 225 \degree
\tan 315 \degree
Hence find the exact value of the following:
\sin 135\degree
\cos 225 \degree
\tan 315 \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. State the exact coordinates of each point:
Q
R
S
Write the following in terms of an equivalent ratio of 60 \degree:
\sin 120\degree
\cos 240 \degree
\tan 300 \degree
Hence find the exact value of the following:
\sin 120\degree
\cos 240 \degree
\tan 300 \degree
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. State the exact coordinates of each point:
Q
R
S
Write the following in terms of an equivalent ratio of 30 \degree:
\cos 150\degree
\sin 210 \degree
\tan 330 \degree
Hence find the exact value of the following:
\cos 150\degree
\sin 210 \degree
\tan 330\degree