Math / Trigonometry

QuestionAnswer the following. (a) Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with $1260^{\circ}$ . (b) Find an angle between 0 and $2 \pi$ that is coterminal with $-\frac{23 \pi}{12}$ .
Give exact values for your answers.

Studdy Solution

STEP 1

What is this asking? We need to find angles within a specific range that point in the same direction as the given angles. Watch out! Remember, coterminal angles land on the same spot, even though they might have spun around a different number of times!

STEP 2

1. Find the coterminal angle between $0^{\circ}$ and $360^{\circ}$ .
2. Find the coterminal angle between 0 and $2\pi$ .

STEP 3

We're given the angle $1260^{\circ}$ and we want to find an angle between $0^{\circ}$ and $360^{\circ}$ that points in the same direction.
A full circle is $360^{\circ}$ , so we can subtract multiples of $360^{\circ}$ until we land in the desired range.
Let's divide $1260^{\circ}$ by $360^{\circ}$ to see how many full rotations we have:
$\frac{1260}{360} = 3.5$

STEP 4

This tells us there are 3 full rotations and an extra half rotation ( $0.5 \cdot 360^{\circ} = 180^{\circ}$ ).
So, if we subtract three full rotations from $1260^{\circ}$ , we'll find our coterminal angle:
$1260^{\circ} - 3 \cdot 360^{\circ} = 1260^{\circ} - 1080^{\circ} = 180^{\circ}$

STEP 5

Now, we have $- \frac{23\pi}{12}$ , a negative angle!
We want an angle between 0 and $2\pi$ that points in the same direction.
Since it's negative, we'll add multiples of $2\pi$ until we're in the right range.
Remember, $2\pi$ represents a full rotation.

STEP 6

Let's start by adding $2\pi$ to our given angle:
$-\frac{23\pi}{12} + 2\pi$ To add these, we need a common denominator.
We can rewrite $2\pi$ as $\frac{24\pi}{12}$ :
$-\frac{23\pi}{12} + \frac{24\pi}{12} = \frac{(-23 + 24)\pi}{12} = \frac{\pi}{12}$ Look at that, we landed right in our desired range of 0 to $2\pi$ !
Awesome!

STEP 7

(a) The angle between $0^{\circ}$ and $360^{\circ}$ coterminal with $1260^{\circ}$ is $180^{\circ}$ .
(b) The angle between 0 and $2\pi$ coterminal with $-\frac{23\pi}{12}$ is $\frac{\pi}{12}$ .

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QuestionAnswer the following. (a) Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with $1260^{\circ}$ . (b) Find an angle between 0 and $2 \pi$ that is coterminal with $-\frac{23 \pi}{12}$ .
Give exact values for your answers.

Studdy Solution

STEP 1

What is this asking? We need to find angles within a specific range that point in the same direction as the given angles. Watch out! Remember, coterminal angles land on the same spot, even though they might have spun around a different number of times!

STEP 2

1. Find the coterminal angle between $0^{\circ}$ and $360^{\circ}$ .
2. Find the coterminal angle between 0 and $2\pi$ .

STEP 3

STEP 4

This tells us there are 3 full rotations and an extra half rotation ( $0.5 \cdot 360^{\circ} = 180^{\circ}$ ).
So, if we subtract three full rotations from $1260^{\circ}$ , we'll find our coterminal angle:
$1260^{\circ} - 3 \cdot 360^{\circ} = 1260^{\circ} - 1080^{\circ} = 180^{\circ}$

STEP 5

Now, we have $- \frac{23\pi}{12}$ , a negative angle!
We want an angle between 0 and $2\pi$ that points in the same direction.
Since it's negative, we'll add multiples of $2\pi$ until we're in the right range.
Remember, $2\pi$ represents a full rotation.

STEP 6

STEP 7

(a) The angle between $0^{\circ}$ and $360^{\circ}$ coterminal with $1260^{\circ}$ is $180^{\circ}$ .
(b) The angle between 0 and $2\pi$ coterminal with $-\frac{23\pi}{12}$ is $\frac{\pi}{12}$ .

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QuestionAnswer the following. (a) Find an angle between 0∘0^{\circ}0∘ and 360∘360^{\circ}360∘ that is coterminal with 1260∘1260^{\circ}1260∘. (b) Find an angle between 0 and 2π2 \pi2π that is coterminal with −23π12-\frac{23 \pi}{12}−1223π​.Give exact values for your answers.

Studdy Solution

STEP 1

What is this asking? We need to find angles within a specific range that point in the same direction as the given angles. Watch out! Remember, coterminal angles land on the same spot, even though they might have spun around a different number of times!

STEP 2

1. Find the coterminal angle between 0∘0^{\circ}0∘ and 360∘360^{\circ}360∘.2. Find the coterminal angle between 0 and 2π2\pi2π.

STEP 3

STEP 4

STEP 5

Now, we have −23π12- \frac{23\pi}{12}−1223π​, a negative angle!We want an angle between 0 and 2π2\pi2π that points in the same direction.Since it's negative, we'll *add* multiples of 2π2\pi2π until we're in the right range.Remember, 2π2\pi2π represents a full rotation.

STEP 6

STEP 7

(a) The angle between 0∘0^{\circ}0∘ and 360∘360^{\circ}360∘ coterminal with 1260∘1260^{\circ}1260∘ is 180∘180^{\circ}180∘.(b) The angle between 0 and 2π2\pi2π coterminal with −23π12-\frac{23\pi}{12}−1223π​ is π12\frac{\pi}{12}12π​.

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QuestionAnswer the following. (a) Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with $1260^{\circ}$ . (b) Find an angle between 0 and $2 \pi$ that is coterminal with $-\frac{23 \pi}{12}$ .
Give exact values for your answers.

1. Find the coterminal angle between $0^{\circ}$ and $360^{\circ}$ .
2. Find the coterminal angle between 0 and $2\pi$ .

Now, we have $- \frac{23\pi}{12}$ , a negative angle!
We want an angle between 0 and $2\pi$ that points in the same direction.
Since it's negative, we'll add multiples of $2\pi$ until we're in the right range.
Remember, $2\pi$ represents a full rotation.

(a) The angle between $0^{\circ}$ and $360^{\circ}$ coterminal with $1260^{\circ}$ is $180^{\circ}$ .
(b) The angle between 0 and $2\pi$ coterminal with $-\frac{23\pi}{12}$ is $\frac{\pi}{12}$ .