Solved Convert each degree measure into radians and each | Chegg.com - Free Printable

Let’s solve each part step by step.

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## Part 1: Convert degrees to radians and radians to degrees

We use the conversion formulas:

- Degrees → Radians:
\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]

- Radians → Degrees:
\[
\text{degrees} = \text{radians} \times \frac{180}{\pi}
\]

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1) 325° → radians

\[
325^\circ \times \frac{\pi}{180} = \frac{325\pi}{180} = \frac{65\pi}{36}
\]

✔ Answer: $\boxed{\dfrac{65\pi}{36}}$

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2) 60° → radians

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

✔ Answer: $\boxed{\dfrac{\pi}{3}}$

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3) $-\dfrac{4\pi}{3}$ → degrees

\[
-\frac{4\pi}{3} \times \frac{180}{\pi} = -4 \times 60 = -240^\circ
\]

✔ Answer: $\boxed{-240^\circ}$

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4) $\dfrac{23\pi}{12}$ → degrees

\[
\frac{23\pi}{12} \times \frac{180}{\pi} = \frac{23 \times 180}{12} = \frac{4140}{12} = 345^\circ
\]

✔ Answer: $\boxed{345^\circ}$

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5) 570° → radians

\[
570^\circ \times \frac{\pi}{180} = \frac{570\pi}{180} = \frac{19\pi}{6}
\]

✔ Answer: $\boxed{\dfrac{19\pi}{6}}$

---

6) -315° → radians

\[
-315^\circ \times \frac{\pi}{180} = -\frac{315\pi}{180} = -\frac{7\pi}{4}
\]

✔ Answer: $\boxed{-\dfrac{7\pi}{4}}$

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## Part 2: Convert decimal degrees ↔ degrees-minutes-seconds (DMS)

Recall:
- 1 degree = 60 minutes (′)
- 1 minute = 60 seconds (″)

To convert decimal degrees to DMS:
1. Whole number = degrees
2. Multiply decimal part by 60 → whole number = minutes
3. Multiply remaining decimal by 60 → seconds

To convert DMS to decimal degrees:
\[
\text{decimal} = \text{degrees} + \frac{\text{minutes}}{60} + \frac{\text{seconds}}{3600}
\]

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7) 128.77° → DMS

- Degrees: 128°
- Decimal: 0.77 × 60 = 46.2 → Minutes: 46′
- Decimal: 0.2 × 60 = 12″

✔ Answer: $\boxed{128^\circ\ 46'\ 12''}$

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8) 232° 7′ 57″ → decimal degrees

\[
232 + \frac{7}{60} + \frac{57}{3600} = 232 + 0.116666... + 0.015833... ≈ 232.1325^\circ
\]

✔ Answer: $\boxed{232.1325^\circ}$

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9) -154° 47′ 42″ → decimal degrees

First, ignore sign, compute positive value, then apply negative.

\[
154 + \frac{47}{60} + \frac{42}{3600} = 154 + 0.783333... + 0.011666... ≈ 154.795^\circ
\]

Apply negative:
✔ Answer: $\boxed{-154.795^\circ}$

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10) -0.9225° → DMS

We’ll work with absolute value first: 0.9225°

- Degrees: 0°
- 0.9225 × 60 = 55.35 → Minutes: 55′
- 0.35 × 60 = 21″

So, 0° 55′ 21″ → Apply negative sign.

✔ Answer: $\boxed{-0^\circ\ 55'\ 21''}$

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## Part 3: Find the measure of each angle (from diagrams)

We need to interpret the diagrams.

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11) Diagram: Angle shown as 30° below positive x-axis, measured clockwise from positive x-axis.

Since it’s measured clockwise (standard position is counterclockwise), this is a negative angle.

But in standard position, we can also express it as a positive coterminal angle:

\[
360^\circ - 30^\circ = 330^\circ
\]

Alternatively, if they want the smallest positive angle, 330°.

If they want the actual rotation shown (clockwise), then -30°.

✔ Answer: $\boxed{330^\circ}$ or $\boxed{-30^\circ}$ — *usually, we give the smallest positive angle unless specified otherwise.*

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12) Diagram: Angle is labeled as $\frac{\pi}{6}$, but it's in the second quadrant, measured from the negative x-axis? Wait — let’s look.

Actually, the diagram shows an angle from the positive x-axis, going counterclockwise, and the label says “$\frac{\pi}{6}$” near the angle, but that seems incorrect because it’s clearly larger than 90°? Wait — no, looking again:

Wait — actually, the diagram shows an angle starting from the positive x-axis, going counterclockwise, and the arc goes into the second quadrant. But the label says “$\frac{\pi}{6}$” — that can’t be right if it’s going past 90°.

Wait — perhaps the angle shown is supplementary to $\frac{\pi}{6}$? Let me re-read.

Actually, in the diagram, the angle marked is between the negative x-axis and the terminal side, which is $\frac{\pi}{6}$. So the total angle from positive x-axis is:

\[
\pi - \frac{\pi}{6} = \frac{5\pi}{6}
\]

✔ Answer: $\boxed{\dfrac{5\pi}{6}}$

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13) Diagram: Angle shown is from positive y-axis, going clockwise to terminal side. The label says “$\frac{\pi}{6}$”.

Standard position: angles are measured from positive x-axis, counterclockwise.

The angle shown is clockwise from positive y-axis — so from positive x-axis, it’s:

Positive y-axis is at $\frac{\pi}{2}$, then we go clockwise $\frac{\pi}{6}$, so:

\[
\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}
\]

But since it’s clockwise from y-axis, and we’re measuring from x-axis counterclockwise, this is still in the first quadrant, so:

✔ Answer: $\boxed{\dfrac{\pi}{3}}$

Wait — actually, if you start at positive x-axis, go counterclockwise to positive y-axis ($\frac{\pi}{2}$), then go clockwise $\frac{\pi}{6}$, you end up at $\frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}$ — yes, that’s correct.

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14) Diagram: Angle shown is $\frac{\pi}{3}$ in the fourth quadrant, measured from positive x-axis clockwise? Or counterclockwise?

Label says “$\frac{\pi}{3}$”, and arrow goes clockwise from positive x-axis.

So, if measured clockwise, it’s negative:

\[
-\frac{\pi}{3}
\]

Or as a positive coterminal angle:

\[
2\pi - \frac{\pi}{3} = \frac{5\pi}{3}
\]

✔ Answer: $\boxed{\dfrac{5\pi}{3}}$ or $\boxed{-\dfrac{\pi}{3}}$ — again, usually smallest positive angle is preferred.

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## ✔ Final Answers Summary:

Part 1:

1) $\boxed{\dfrac{65\pi}{36}}$
2) $\boxed{\dfrac{\pi}{3}}$
3) $\boxed{-240^\circ}$
4) $\boxed{345^\circ}$
5) $\boxed{\dfrac{19\pi}{6}}$
6) $\boxed{-\dfrac{7\pi}{4}}$

Part 2:

7) $\boxed{128^\circ\ 46'\ 12''}$
8) $\boxed{232.1325^\circ}$
9) $\boxed{-154.795^\circ}$
10) $\boxed{-0^\circ\ 55'\ 21''}$

Part 3:

11) $\boxed{330^\circ}$
12) $\boxed{\dfrac{5\pi}{6}}$
13) $\boxed{\dfrac{\pi}{3}}$
14) $\boxed{\dfrac{5\pi}{3}}$

*(Note: For angles 11 and 14, negative equivalents are also mathematically valid, but positive coterminal angles are typically expected unless otherwise specified.)*