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Trigonometry : Unit Circle

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Example Question #1 :Unit Circle

What point corresponds to the angle $-\pi$ on the unit circle?

Possible Answers:

(1,0)

(1,1)

(-1,0)

(-1,-1)

(0,0)

Correct answer:

(-1,0)

Explanation:

The unit circle is the circle of radius one centered at the origin (0,0) in the Cartesian coordinate system. $-\pi$ 相当于 $-180^{\circ}$ which corresponds to the point (-1,0) on the unit circle.

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Example Question #2 :Unit Circle

What point corresponds to an angle of $\frac{5\pi}{2}$ radians on the unit circle?

Possible Answers:

(0,1)

(-1,-1)

(-1,1)

(0,0)

(1,1)

Correct answer:

(0,1)

Explanation:

The unit circle is the circle of radius one centered at the origin (0,0) in the Cartesian coordinate system. $\frac{5\pi}{2}$ radians is equivalent to $\frac{5\pi}{2}\times \frac{180^{\circ}}{\pi}=450^{\circ}$ . This is a full circle $360^{\circ}$ plus a quarter-turn $90^{\circ}$ more. So, the angle $\frac{5\pi}{2}$ corresponds to the point (0,1) on the unit circle.

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Example Question #1 :Unit Circle

Give the angle between $0^{\circ}$ and $360^{\circ}$ that corresponds to the point (-1,0) .

Possible Answers:

$180^{\circ}$

$120^{\circ}$

$90^{\circ}$

$45^{\circ}$

$60^{\circ}$

Correct answer:

$180^{\circ}$

Explanation:

The angle $\pi$ or $180^{\circ}$ corresponds to the point (-1,0) .

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Example Question #1 :Unit Circle

What point corresponds to the angle $\frac{\pi}{2}$ on the unit circle?

Possible Answers:

(0,-1)

(1,1)

(-1,-1)

(1,0)

(0,1)

Correct answer:

(0,1)

Explanation:

The unit circle is the circle of radius one centered at the origin (0,0) in the Cartesian coordinate system. $\frac{\pi}{2}$ 相当于 $90^{\circ}$ which corresponds to the point (0,1) on the unit circle.

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Example Question #1 :Unit Circle And Radians

Which of the following points is NOT on the unit circle?

Possible Answers:

(1,0)

$(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$

$(\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2})$

$(\frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

(0, -1)

Correct answer:

$(\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2})$

Explanation:

For a point to be on the unit circle, it has to have a radius of one. Therefore, the sum of the squares of point's coordinates must also equal one.

Let's try the point $(\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2})$ .

$(\frac{\sqrt{3}}{2})^{2} + (\frac{\sqrt{2}}{2})^{2} = \frac{5}{4} \neq 1$

Therefore, this point is not on the unit circle.

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Example Question #6 :Unit Circle

When looking at the unit circle, what are the coordinates for an angle of $\frac{\pi}{4}$ ?

Possible Answers:

$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$

$\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

(0,1)

$(\sqrt{2},\sqrt{2})$

$\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$

Correct answer:

$\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

Explanation:

The coordinates of the point on the circle for each angle are $(\cos\theta,\sin\theta)$ .

Since $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ , the point will be $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ .

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Example Question #1 :Unit Circle And Radians

What is the radius of the unit circle?

Possible Answers:

$\frac{3}{4}$

$\frac{1}{2}$

Correct answer:

Explanation:

By definition, the radius of the unit circle is 1.

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Example Question #1 :Unit Circle And Radians

What point corresponds to the angle $\frac{\pi}{3}$ on the unit circle?

Possible Answers:

(1,1)

$\left(\frac{1}{2},\frac{1}{2}\right)$

$\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$

(0,-1)

$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$

Correct answer:

$\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$

Explanation:

In order to find the point corresponds to the angle $\frac{\pi}{3}$ on the unit circle we can write:

$t=\frac{\pi}{3}\ or\ 60^{\circ}$

In the unit circle which has the radious of R=1 we can write:

$cos\ t=\frac{x}{R}=\frac{x}{1}=x\Rightarrow x=cos(60^{\circ})=\frac{1}{2}$

$sin\ t=\frac{y}{R}=\frac{y}{1}=y\Rightarrow y=sin(60^{\circ})=\frac{\sqrt{3}}{2}$

So the point corresponds to the angle $\frac{\pi}{3}$ on the unit circle is $(\frac{1}{2},\frac{\sqrt{3}}{2})$

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Example Question #1 :Unit Circle

What point corresponds to an angle of $540^{\circ}$ on the unit circle?

Possible Answers:

(-1,-1)

(1,-1)

(0,1)

(-1,0)

(1,1)

Correct answer:

(-1,0)

Explanation:

The unit circle is the circle of radius one centered at the origin (0,0) in the Cartesian coordinate system. $540^{\circ}$ is a full circle $360^{\circ}$ plus a $180^{\circ}$ more. So, the angle $540^{\circ}$ corresponds to the point (-1,0) on the unit circle.

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Example Question #1 :Unit Circle

What must be the area of the unit circle?

Possible Answers:

$\frac{\pi}{2}$

$\pi$

$2\pi$

$\frac{\pi}{4}$

The area can vary.

Correct answer:

$\pi$

Explanation:

The unit circle must have a radius of 1.

Use the circular area formula to find the area.

$A_{circle}= \pi r^2= \pi$

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Trigonometry : Unit Circle

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 :Unit Circle

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