8.05 Trigonometric functions on the unit circle
Interactive practice questions
The diagram shows a unit circle with point $P\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$P(1√2,1√2) marked on the circle. Point $P$P represents a rotation of $45^\circ$45° counterclockwise around the origin from the positive $x$x-axis.
Complete the table of values.
| $\sin45^\circ$sin45° | $\cos45^\circ$cos45° | $\tan45^\circ$tan45° |
| $\frac{\editable{}}{\sqrt{\editable{}}}$√ | $\frac{\editable{}}{\sqrt{\editable{}}}$√ | $\editable{}$ |
On the coordinate axes below, a $45^\circ$45° angle has also been marked in the second, third, and fourth quadrants. For each quadrant, what is the relative angle (the counterclockwise rotation from the positive $x$x-axis)?
| Quadrant | $I$I | $II$II | $III$III | $IV$IV |
|---|---|---|---|---|
| Relative angle | $45^\circ$45° | $\editable{}$$^\circ$° | $\editable{}$$^\circ$° | $\editable{}$$^\circ$° |
Points $Q$Q, $R$R and $S$S mark rotations of point $P$P to the corresponding angles on the unit circle.
Complete the coordinates of each point.
$Q$Q$\left(\frac{\editable{}}{\sqrt{\editable{}}},\frac{\editable{}}{\sqrt{\editable{}}}\right)$(√,√)
$R$R$\left(\frac{\editable{}}{\sqrt{\editable{}}},\frac{\editable{}}{\sqrt{\editable{}}}\right)$(√,√)
$S$S$\left(\frac{\editable{}}{\sqrt{\editable{}}},\frac{\editable{}}{\sqrt{\editable{}}}\right)$(√,√)
Complete the gaps using any of the following numbers: $1,2,3,4$1,2,3,4
$\sin x$sinx is positive in quadrants $\editable{},\editable{}$,.
$\cos x$cosx is positive in quadrants $\editable{},\editable{}$,.
$\tan x$tanx is positive in quadrants $\editable{},\editable{}$,.
To find $\sin173^\circ$sin173°, we can first find the sine ratio of which acute angle?
Consider the graph of the unit circle shown below.
Consider the graph of the unit circle shown below.
Use the figure to find the value of $\sin\left(\frac{7\pi}{6}\right)$sin(7π6).