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4.05 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(12,12) marked on the circle. Point $P$P represents a rotation of $45^\circ$45° counterclockwise around the origin from the positive $x$x-axis.

a

Complete the table of values.

$\sin45^\circ$sin45° $\cos45^\circ$cos45° $\tan45^\circ$tan45°
$\frac{\editable{}}{\sqrt{\editable{}}}$ $\frac{\editable{}}{\sqrt{\editable{}}}$ $\editable{}$
b

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$°
c

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$\left(\frac{\editable{}}{\sqrt{\editable{}}},\frac{\editable{}}{\sqrt{\editable{}}}\right)$(,)

d

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 $\editable{},\editable{}$,.

e

To find $\sin173^\circ$sin173°, we can first find the sine ratio of which acute angle?

Easy
8min

Consider the graph of the unit circle shown below.

Easy
< 1min

Consider the graph of the unit circle shown below.

Easy
< 1min

Use the figure to find the value of $\sin\left(\frac{7\pi}{6}\right)$sin(7π6).

Easy
< 1min
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