Suppose that we have an angle in the coordinate plane positioned in the usual way. Its reference angle, denoted is the smallest positive acute angle formed with the -axis.
For example, let's consider the angles and These angles lie in the first, second, third, and fourth quadrants, respectively. And each of these angles has a reference angle as shown below.
Note: Any reference angle always lies between and inclusively:
The diagram above shows the angle in the coordinate plane. What is its reference angle
The reference angle, denoted is the angle that makes with the -axis.
Since is in the second quadrant, its reference angle is given by
The diagram above shows the angle $\theta = 70^\circ$ in the coordinate plane. What is its reference angle $\theta_R ?$
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If the angle $\theta = 177^\circ$ is represented in the coordinate plane in the usual way, then what is its reference angle $\theta_R?$
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a
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$5^\circ$ |
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b
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$77^\circ$ |
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c
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$87^\circ$ |
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d
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$3^\circ$ |
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e
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$17^\circ$ |
The diagram above shows the angle $\theta = 210^\circ$ in the coordinate plane. What is its reference angle $\theta_R?$
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a
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$60^\circ$ |
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b
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$70^\circ$ |
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c
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$20^\circ$ |
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d
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$10^\circ$ |
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e
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$30^\circ$ |
If the angle is represented in the coordinate plane in the usual way, then what is its reference angle?
To help us to visualize the angle, we first convert the measure of the angle into degrees.
To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by This gives
Therefore is equivalent to which lies in the third quadrant.
The reference angle, denoted is the angle that makes with the -axis.
As is in the third quadrant, its reference angle is given by
Therefore,
The diagram above shows the angle $\theta = \dfrac{7\pi}{4}$ in the coordinate plane. What is its reference angle $\theta_R?$
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a
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$\dfrac{3\pi}{8}$ |
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b
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$\dfrac{\pi}{4}$ |
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c
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$\dfrac{\pi}{3}$ |
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d
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$\dfrac{\pi}{8}$ |
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e
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$\dfrac{\pi}{6}$ |
The angle $\theta = \dfrac{7\pi}{5}$ is represented in the coordinate plane in the usual way. Expressed in radians, what is its reference angle $\theta_R?$
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b
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The diagram above shows a unit circle and the angle is measured in the usual way. Given that the -coordinate of the point is calculate the reference angle in degrees to the nearest integer.
First, we create a right triangle by drawing a vertical line from to the -axis.
So we have the following triangle, where is the reference angle of
We know that the hypotenuse has a length of because it is a radius of the unit circle.
Now, using the triangle above, we have
Therefore,
The diagram above shows a unit circle and the angle $\theta$ measured in the usual way. Given that the $y$-coordinate of the point $P$ is $0.91,$ calculate the reference angle $\theta_R$ to the nearest integer.
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b
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The diagram above shows a unit circle and the angle $\theta$ measured in the usual way. Given that the $y$-coordinate of the point $P$ is $-0.81,$ calculate the reference angle $\theta_R$ to the nearest integer.
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a
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$58^\circ$ |
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b
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$60^\circ$ |
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c
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$54^\circ$ |
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d
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$49^\circ$ |
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e
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$51^\circ$ |
An angle in the coordinate plane (measured in the usual way) has a reference angle Which of the following cannot be
Let's remind ourselves of the relationship between and
Using let's calculate all possible values of
If lies in the first quadrant, then
If lies in the second quadrant, then
If lies in the third quadrant, then
If lies in the fourth quadrant, then
Therefore, cannot be equal to
An angle $\theta$ in the coordinate plane (measured in the usual way) has a reference angle $\theta_R = 60^\circ.$ Which of the following cannot be $\theta?$
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a
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$260^\circ$ |
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b
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$120^\circ$ |
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c
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$300^\circ$ |
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d
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$240^\circ$ |
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e
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$60^\circ$ |
An angle $\theta$ in the coordinate plane (measured in the usual way) has a reference angle $\theta_R = \dfrac{\pi}{8}.$ List, in ascending order, all the possible values of $\theta$ in radian in the range $0^\circ \leq \theta < 2\pi$
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a
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b
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c
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d
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e
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