Let's consider the unit circle with a central angle in the first quadrant, measured in the usual way.
Notice that
the -coordinate of coincides with the cosine of the angle and
the -coordinate coincides with the sine of the angle
So, we have
Consequently, if we know the angle we can easily work out the coordinates of the point and vice-versa. This works regardless of whether the angle is given in degrees or radians.
For now, we'll focus on points located in the first quadrant. Later, we will encounter points in all quadrants.
The diagram above shows the unit circle. What are the coordinates of the point
Any point on the unit circle is related to the central angle as follows:
We're given that at the point Therefore, we have
So, the coordinates of are
The diagram above shows the unit circle. What is the $x$-coordinate of the point $P?$
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a
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$\dfrac{4}{5}$ |
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b
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$\dfrac{1}{2}$ |
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c
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$\dfrac{\sqrt 3}{2}$ |
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d
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$\dfrac{3}{5}$ |
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e
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$\dfrac{\sqrt 3}{3}$ |
The diagram above shows a unit circle. What is the $y$-coordinate of the point $P?$
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a
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$\dfrac{\sqrt 3}{3}$ |
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b
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$\sqrt 3$ |
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c
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$\dfrac{1}{4}$ |
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d
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$\dfrac{1}{2}$ |
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e
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$\dfrac{\sqrt 3}{2}$ |
The diagram above shows a unit circle. Rounding to two significant figures, the $y$-coordinate of the point $P$ is
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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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Let's consider the unit circle and a point that lies on the circle.
To prove that and let's examine the triangle that's formed by and the -axis.
Notice that:
The hypotenuse has length since it is a radius of the unit circle.
The adjacent and opposite sides have lengths and respectively.
Therefore, using the definitions of and , we have
The diagram above shows the unit circle. Given that the -coordinate of the point is what is the measure of the angle
We're given that at the point Therefore, we have
We compute the angle using the inverse sine:
The diagram above shows the unit circle. The $x$-coordinate of the point $P$ is $0.4.$ Rounded to two decimal places, what is the measure of the angle $\theta?$
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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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The diagram above shows the unit circle. Given that the $x$-coordinate of the point $P$ is $\dfrac{\sqrt 2}{2},$ what is the measure of the angle $\theta?$
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a
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$30^\circ$ |
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b
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$45^\circ$ |
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c
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$40^\circ$ |
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d
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$75^\circ$ |
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e
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$50^\circ$ |
The point lies on the unit circle, as shown. What is
We're given that at the point Therefore, we have
Knowing that we can calculate as follows:
The point $P\left( \dfrac{\sqrt 6}{4},\dfrac{\sqrt {10}}{4}\right)$ lies on the unit circle, as shown. What is $\sec\theta?$ Do not round your answer.
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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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The point $P\left( \dfrac{\sqrt 3}{2},\dfrac{1}{2}\right)$ lies on the unit circle, as shown. What is $\csc\theta?$
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a
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$1$ |
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b
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$\dfrac{\sqrt 3}{2}$ |
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c
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$\dfrac{2\sqrt 3}{3}$ |
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d
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$2$ |
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e
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$\dfrac{1}{2}$ |
To find the tangent of an angle using the unit circle, we need to use the relationship
To see why this is true, let's take a look at the following triangle.
For this particular triangle, we have
When we take and divide it by we get
Although this proof is only for acute angles, the identity is true for any angle, provided that is well-defined.
Finally, since it immediately follows that
The point lies on the unit circle, as shown. What is
We're given that and at the point Therefore, we have
Knowing that we can calculate as follows:
The point $P\left( \dfrac{1}{2},\dfrac{\sqrt 3}{2}\right)$ lies on the unit circle, as shown. What is $\tan\theta?$
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a
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$\sqrt 2$ |
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b
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$\dfrac{\sqrt 3}{3}$ |
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c
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$\dfrac{\sqrt 2}{2}$ |
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d
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$\sqrt 3$ |
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e
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$\dfrac{\sqrt 3}{2}$ |
The point $P\left( \dfrac{2}{7},\dfrac{3\sqrt{5}}{7}\right)$ lies on the unit circle, as shown. What is $\cot\theta?$ Do not round your answer.
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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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