Unit Circle - Trigonometry

Card 0 of 14

Question

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

Answer

The unit circle is the circle of radius one centered at the origin 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 on the unit circle.

Compare your answer with the correct one above

Question

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

Answer

The unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. $-\pi$ is equivalent to $-180^{\circ}$ which corresponds to the point on the unit circle.

Compare your answer with the correct one above

Question

Give the angle between $0^{\circ}$ and $360^{\circ}$ that corresponds to the point .

Answer

The angle $\pi$ or $180^{\circ}$ corresponds to the point .

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

The unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. $\frac{\pi}{2}$ is equivalent to $90^{\circ}$ which corresponds to the point on the unit circle.

Compare your answer with the correct one above

Question

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

Answer

The unit circle is the circle of radius one centered at the origin 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 on the unit circle.

Compare your answer with the correct one above

Question

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

Answer

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

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

Therefore, this point is not on the unit circle.

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

What is the radius of the unit circle?

Answer

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

Compare your answer with the correct one above

Question

What must be the area of the unit circle?

Answer

The unit circle must have a radius of 1.

Use the circular area formula to find the area.

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

Compare your answer with the correct one above

Question

Suppose there is exists an angle, $\theta = \frac{n\pi }{x}{}$ such that $cos \ \theta = \frac{1}{2}$ .

For what values of and make this trigonometric ratio possible?

Answer

The only values such that

$cos \ \theta = \frac{1}{2}$

are at the values:

$\theta = \frac{\pi}{3} , \frac{5 \pi }{3}$

This means that the only choice for is . or $n = \ 5$ achieve the necessary angles to satisfy this trigonometric ratio.

Compare your answer with the correct one above

Question

If $tan\left ( \theta \right )=\frac{3}{4}$ , and $\pi \leq \theta \leq \frac{3\pi}{2}$ , what is $sin(\theta)$ ?

Answer

The tangent of an angle yields the ratio of the opposite side to the adjacent side.

If this ratio is $\frac{3}{4}$ , we can see that this is a Pythagorean triple (3-4-5); the absolute value of the sine of this angle would be $\frac{3}{5}$ .

However, the question indicates that this angle lies in the 3rd quadrant. The sine of any angle in the 3rd or 4th quadrant is negative, since it is equivalent to the y-coordinate of the corresponding point on the unit circle.

Therefore,

$sin(\theta)=-\frac{3}{5}$ .

Compare your answer with the correct one above

Question

If $cos(\phi)=-\frac{1}{2}$ , which of the following angles is NOT a possible value for $\phi$ ?

Answer

On the unit circle, the cosine of an angle yields the x-coordinate.

There are two angles at which the _x-_coordinate on the unit circle is $-\frac{1}{2}$ : $120^{\circ}$ and $240^{\circ}$ .

$-120^{\circ}$ is coterminal with $240^{\circ}$ , and $480^{\circ}$ is coterminal with $120^{\circ}$ .

$300^{\circ}$ is in the 4th quadrant, and has a positive x-coordinate.

Compare your answer with the correct one above

Question

How many degrees are in a unit circle?

Answer

Step 1: Define a Unit Circle:

A unit circle is used in Trigonometry to draw and describe distinct angles. The unit circle works along with the coordinate grid.

Step 2: There are quadrants in the coordinate grid, each quadrant can fit degrees.

Step 3: Multiply how many degrees in quadrant by .. to get the full unit circle:

$90 \cdot 4=360$

Compare your answer with the correct one above

Tap the card to reveal the answer