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Math  /  Trigonometry

Question(a) Convert 540∘540^{\circ} to radian measure in terms of π\pi. □\square radians (b) Convert −13π6-\frac{13 \pi}{6} radians to degree measure. □\square

Studdy Solution

STEP 1

What is this asking? We need to switch between two ways of measuring angles: degrees and radians, going both ways! Watch out! Remember that a full circle is 360360 degrees, which is the same as 2π2\pi radians.
Don't mix them up!

STEP 2

1. Convert Degrees to Radians
2. Convert Radians to Degrees

STEP 3

Alright, we're starting with 540540 **degrees** and want to turn that into **radians**.
The key is to remember how degrees and radians relate to a full circle.

STEP 4

A full circle is 360360 degrees, which is the same as 2π2\pi radians.
This means 360∘=2π360^\circ = 2\pi radians.
We can write that as a **conversion factor**: 2π radians360∘=1\frac{2\pi \text{ radians}}{360^\circ} = 1.
Since multiplying by one doesn't change a value, we can use this to convert!

STEP 5

So, we **multiply** our 540540 degrees by this fancy 11: 540∘⋅2π radians360∘540^\circ \cdot \frac{2\pi \text{ radians}}{360^\circ}

STEP 6

Notice how the **degree symbols cancel out** (we're dividing degrees by degrees, which equals one).
We're left with: 540⋅2π360 radians\frac{540 \cdot 2\pi}{360} \text{ radians}

STEP 7

Now, we **simplify**! 540540 divided by 360360 simplifies to 540360=32\frac{540}{360} = \frac{3}{2}.
So, we get: 32⋅2π radians\frac{3}{2} \cdot 2\pi \text{ radians}

STEP 8

The twos **cancel out** (we're multiplying by two and dividing by two, which equals one), leaving us with our **final answer**: 3π3\pi radians!

STEP 9

Now, we're going the other way!
We have −13π6-\frac{13\pi}{6} radians and we want to convert it to **degrees**.

STEP 10

We can use our **conversion factor** again, but flipped this time: 360∘2π radians=1\frac{360^\circ}{2\pi \text{ radians}} = 1.

STEP 11

Let's **multiply** our radians by this new form of 11: −13π6 radians⋅360∘2π radians-\frac{13\pi}{6} \text{ radians} \cdot \frac{360^\circ}{2\pi \text{ radians}}

STEP 12

The **radian units cancel out** (we're dividing radians by radians, which equals one), and the π\pi also **cancels out** (we're multiplying by π\pi and dividing by π\pi, which equals one).
We get: −136⋅360∘2-\frac{13}{6} \cdot \frac{360^\circ}{2}

STEP 13

Let's **simplify**! 360360 divided by 22 is 180180, so we have: −136⋅180∘-\frac{13}{6} \cdot 180^\circ

STEP 14

180180 divided by 66 is 3030, so we have: −13⋅30∘-13 \cdot 30^\circ

STEP 15

Finally, −13-13 times 3030 is −390-390, giving us our **final answer**: −390∘-390^\circ!

STEP 16

(a) 3π3\pi radians (b) −390∘-390^\circ

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