Unit Circle and Radians - Trigonometry

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Question

What is the value of ?

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Answer

To help with this one, draw a 45-45-90 triangle. With legs equal to 1 and a hypotenuse of $\sqrt2$ .

Then, use the definition of tangent as opposite over adjacent to find the value.

Since the legs are congruent, we get that the ratio is 1.

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Question

What is ?

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Answer

Recall that on the unit circle, sine represents the y coordinate of the unit circle.

Then, since we are at 90˚, we are at the positive y axis, the point (0,1).

At this point on the unit circle, the y value is 1.

Thus .

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Question

Which of the following is NOT a special angle on the unit circle?

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Answer

For an angle to be considered a special angle, the angle must be able to produce a or a triangle.

The only angle that is not capable of the special angles formation is $75 \textup{ degrees}$ .

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Question

What are the ways to write 360o and 720o in radians?

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Answer

$\small 360^o=2\pi$ on the unit circle.

$\small \small \small 720^o=4\pi$ on the unit circle.

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Question

Which of the following is not an angle in the unit circle?

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Answer

The unit circle is in two increments: $\frac{\pi}{6}$ : $(\frac{\pi}{6},\frac{2\pi}{6},\frac{3\pi}{6})$ , etc. and $\frac{\pi}{4}$ : $(\frac{\pi}{4},\frac{2\pi}{4},\frac{3\pi}{4})$ , etc. The only answer choice that is not a multiple of either $\frac{\pi}{6}$ or $\frac{\pi}{4}$ is $\frac{\pi}{5}$ .

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Question

What is the equivalent of $72^{\circ}$ , in radians?

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Answer

To convert from degrees to radians, use the equality

$2\pi = 360^{\circ}$ or $\pi = 180^{\circ}$ .

From here we use $\frac{\pi}{180^{\circ}}$ as a unit multiplier to convert our degrees into radians.

$\left (72^{\circ} \right )\left ( \frac{\pi }{180^{\circ}} \right ) = \frac{2\pi}{5}$

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Question

If $0\leq x\leq\pi$ and $cos(5x)=\frac{-1}{2}$ , then =

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Answer

We first make the substitution .

In the interval $0\leq k\leq \pi$ , the equation $\cos k=\frac{-1}{2}$ has the solution $k=\frac{2\pi}{3}$ .

Solving for ,

$\begin{align} 5x&=\frac{2\pi}{3}\ x&=\frac{2\pi}{15} \end{align}$ .

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Question

What is the value of $\sin(\pi/6)$ from the unit circle?

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Answer

From the unit circle, the value of

$\sin(\pi/6)=1/2$ .

This can be found using the coordinate pair associated with the angle $\pi/6$ which is $(\sqrt3/2,1/2)$ .

Recall that the pair are $(\cos,\sin)$ .

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Question

What is

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

using the unit circle?

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Answer

Recall that the unit circle can be broken down into four quadrants. Each quadrant has similar coordinate pairs basic on the angle. The only difference between the actual coordinate pairs is the sign on them. In quadrant I all signs are positive. In quadrant II only sine and cosecant are positive. In quadrant III tangent and cotangent are positive and quadrant IV only cosine and secant are positive.

$\frac{11\pi}{6}$ has the reference angle of $\frac{\pi}{6}$ and lies in quadrant IV therefore .

From the unit circle, the coordinate point of

$(\sqrt3/2,-1/2)$ corresponds with the angle $\frac{11\pi}{6}$ .

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Question

Give the exact value.

Use the unit circle to find:

$\sin\frac{5\pi}{6}$

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Answer

Locate $\frac{5\pi}{6}$ on the unit circle.

Sine is related to the why y value of the coordinate point because it is opposite/Hyp.

In other words the pair of the point located on the unit circle that extends from the origin is $(\cos(\theta),\sin(\theta))$ .

The coordinate pair for $\frac{5\pi}{6}$ is $\left(\frac{-\sqrt{3}}{2},\frac{1}{2}\right)$ thus,

$\sin\frac{5\pi}{6}=\frac{1}{2}$

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Question

Give the exact value.

Use the unit circle to find:

$\cos \frac{4\pi}{3}$

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Answer

Locate $\frac{4\pi}{3}$ on the unit circle.

Cosine is related to the why x value of the points, because it is Adj/Hyp.

In other words the pair of the point located on the unit circle that extends from the origin is $(\cos(\theta),\sin(\theta))$ .

The coordinate point of $\frac{4\pi}{3}$ is $\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right )$ thus,

$\cos \frac{4\pi}{3}=-\frac{1}{2}$ .

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Question

Give the exact value.

Consider the unit circle. What is the value of the given trigonometric function?

$\tan \frac{2\pi}{3}$

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Answer

Use the unit circle to locate the following:

$\frac{2\pi}{3}$

Recall that the pair of the point located on the unit circle that extends from the origin is the following:

$(\cos(\theta),\sin(\theta))$

Since the tangent is equal to the sine divided by the cosine, we need to find the following:

$\sin \frac{2\pi}{3}=\frac{\sqrt{3}}{2}$

$\cos \frac{2\pi}{3}=-\frac{1}{2}$

Now, we will use this information to solve the problem.

$\tan \frac{2\pi}{3}=\frac{\sin\frac{2\pi}{3}}{\cos \frac{2\pi}{3}}$

Substitute the calculated values:

$\frac{\sin\frac{2\pi}{3}}{\cos \frac{2\pi}{3}}=\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}$

Solve. Remember that dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\frac{\sqrt{3}}{2}\times-\frac{2}{1}$

$\frac{\sqrt{3}}{\not{2}}\times-\frac{\not{2}}{1}=-\sqrt{3}$

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Question

Change $\frac{4\pi }{3}$ angle to degrees.

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Answer

In order to change an angle into degrees, you must multiply the radian by $\frac{180^{\circ}}{\pi}$ .

Therefore, to solve:

$\frac{4\pi }{3} \cdot \frac{180^{\circ} }{\pi} = 240^{\circ}$

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Question

Change $\frac{\pi}{5}$ angle to degrees.

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Answer

In order to change radians to degrees, we need to multiply the radian agle measure by $\frac{180^{\circ}}{\pi}$ .

$\frac{\pi}{5}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{5}=36^{\circ}$

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Question

An angle of 40 radians is equal to how many degrees?

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Answer

We know that one radian is equal to $\frac{180^{\circ}}{\pi}$ , so in order to change radians to degrees we need to multiply by $\frac{180^{\circ}}{\pi}$ .

$40\times \frac{180^{\circ}}{\pi}=\frac{7200^{\circ}}{\pi}$

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Question

Change $\frac{17\pi}{5}$ angle to degrees.

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Answer

In order to change radians to degrees we need to multiply the radians by $\frac{180^{\circ}}{\pi}$ .

$\frac{17\pi}{5}\times \frac{180^{\circ}}{\pi}=\frac{17\times 180^{\circ}}{5}=612^{\circ}$

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Question

Change $75^{\circ}$ to radians.

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Answer

In order to change degrees to radians we need to multiply the degrees by $\frac{\pi}{180^{\circ}}$ .

$75^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{5\pi}{12}$

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Question

Change the following expression to degrees:

$\frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}$

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Answer

First we need to simplify the expression:

$\frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}=\frac{\frac{2\pi-\pi}{6}}{\frac{2+1}{6}}=\frac{\pi}{3}$

Now multiply by $\frac{180^{\circ}}{\pi}$ :

$\frac{\pi}{3}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{3}=60^{\circ}$

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Question

Change the following expression to degrees:

${\frac{\pi}{4}-\frac{\pi}{12}}$

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Answer

First we need to simplify the expression:

${\frac{\pi}{4}-\frac{\pi}{12}}=\frac{3\pi-\pi}{12}=\frac{2\pi}{12}=\frac{\pi}{6}$

Then multiply the radians by $\frac{180^{\circ}}{\pi}$ :

$\frac{\pi}{6}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{6}=30^{\circ}$

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Question

Convert the following expression to radians:

$\frac{45^{\circ}+30^{\circ}}{45-30}$

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Answer

First we need to simplify the expression:

$\frac{45^{\circ}+30^{\circ}}{45-30}=\frac{75^{\circ}}{15}=5^{\circ}$

In order to change degrees to radians, we need to multiply by $\frac{\pi}{180^{\circ}}$ :

$5^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{36}$

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