Math / Trigonometry

QuestionFind the reference angle. A) $\frac{11 \pi}{36}$ B) $\frac{\pi}{9}$ C) $\frac{5 \pi}{18}$ D) $\frac{2 \pi}{9}$

Studdy Solution

STEP 1

What is this asking? We need to find the smallest angle between the terminal side of $\frac{41\pi}{18}$ and the x-axis. Watch out! Reference angles are always positive and less than or equal to $\frac{\pi}{2}$ .

STEP 2

1. Revolutions
2. Find the coterminal angle
3. Determine the quadrant and calculate the reference angle

STEP 3

Let's rewrite our angle $\frac{41\pi}{18}$ in terms of full revolutions ( $2\pi$ ).
We can think of $\frac{41\pi}{18}$ as a mixed number of revolutions!

STEP 4

We divide $\frac{41}{18}$ to get $2$ with a remainder of $5$ .
So, $\frac{41\pi}{18}$ is the same as $2\pi + \frac{5\pi}{18}$ .
This means we have two full revolutions and an extra $\frac{5\pi}{18}$ radians!

STEP 5

Coterminal angles are angles that share the same terminal side.
Since $\frac{41\pi}{18}$ is the same as two full revolutions plus $\frac{5\pi}{18}$ , the coterminal angle we're interested in is just $\frac{5\pi}{18}$ .

STEP 6

Since $\frac{5\pi}{18}$ is between $0$ and $\frac{\pi}{2}$ (which is $\frac{9\pi}{18}$ ), our angle is in the first quadrant!

STEP 7

For angles in the first quadrant, the reference angle is simply the angle itself.
Therefore, the reference angle is $\frac{5\pi}{18}$ .

STEP 8

The reference angle for $\frac{41\pi}{18}$ is $\frac{5\pi}{18}$ , which is option C!

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QuestionFind the reference angle. A) $\frac{11 \pi}{36}$ B) $\frac{\pi}{9}$ C) $\frac{5 \pi}{18}$ D) $\frac{2 \pi}{9}$

Studdy Solution

STEP 1

What is this asking? We need to find the smallest angle between the terminal side of $\frac{41\pi}{18}$ and the x-axis. Watch out! Reference angles are always positive and less than or equal to $\frac{\pi}{2}$ .

STEP 2

1. Revolutions
2. Find the coterminal angle
3. Determine the quadrant and calculate the reference angle

STEP 3

Let's rewrite our angle $\frac{41\pi}{18}$ in terms of full revolutions ( $2\pi$ ).
We can think of $\frac{41\pi}{18}$ as a mixed number of revolutions!

STEP 4

We divide $\frac{41}{18}$ to get $2$ with a remainder of $5$ .
So, $\frac{41\pi}{18}$ is the same as $2\pi + \frac{5\pi}{18}$ .
This means we have two full revolutions and an extra $\frac{5\pi}{18}$ radians!

STEP 5

Coterminal angles are angles that share the same terminal side.
Since $\frac{41\pi}{18}$ is the same as two full revolutions plus $\frac{5\pi}{18}$ , the coterminal angle we're interested in is just $\frac{5\pi}{18}$ .

STEP 6

Since $\frac{5\pi}{18}$ is between $0$ and $\frac{\pi}{2}$ (which is $\frac{9\pi}{18}$ ), our angle is in the first quadrant!

STEP 7

For angles in the first quadrant, the reference angle is simply the angle itself.
Therefore, the reference angle is $\frac{5\pi}{18}$ .

STEP 8

The reference angle for $\frac{41\pi}{18}$ is $\frac{5\pi}{18}$ , which is option C!

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QuestionFind the reference angle. A) 11π36\frac{11 \pi}{36}3611π​ B) π9\frac{\pi}{9}9π​ C) 5π18\frac{5 \pi}{18}185π​ D) 2π9\frac{2 \pi}{9}92π​

Studdy Solution

STEP 1

What is this asking? We need to find the smallest angle between the terminal side of 41π18 \frac{41\pi}{18} 1841π​ and the x-axis. Watch out! Reference angles are always positive and less than or equal to π2 \frac{\pi}{2} 2π​.

STEP 2

1. Revolutions2. Find the coterminal angle3. Determine the quadrant and calculate the reference angle

STEP 3

Let's **rewrite** our angle 41π18 \frac{41\pi}{18} 1841π​ in terms of full revolutions (2π 2\pi 2π).We can think of 41π18 \frac{41\pi}{18} 1841π​ as a mixed number of revolutions!

STEP 4

We **divide** 4118 \frac{41}{18} 1841​ to get 2 2 2 with a remainder of 5 5 5.So, 41π18 \frac{41\pi}{18} 1841π​ is the same as 2π+5π18 2\pi + \frac{5\pi}{18} 2π+185π​.This means we have **two full revolutions** and an extra 5π18 \frac{5\pi}{18} 185π​ radians!

STEP 5

Coterminal angles are angles that share the same terminal side.Since 41π18 \frac{41\pi}{18} 1841π​ is the same as two full revolutions plus 5π18 \frac{5\pi}{18} 185π​, the **coterminal angle** we're interested in is just 5π18 \frac{5\pi}{18} 185π​.

STEP 6

Since 5π18 \frac{5\pi}{18} 185π​ is between 0 0 0 and π2 \frac{\pi}{2} 2π​ (which is 9π18 \frac{9\pi}{18} 189π​), our angle is in the **first quadrant**!

STEP 7

For angles in the first quadrant, the reference angle is simply the angle itself.Therefore, the **reference angle** is 5π18 \frac{5\pi}{18} 185π​.

STEP 8

The reference angle for 41π18 \frac{41\pi}{18} 1841π​ is 5π18 \frac{5\pi}{18} 185π​, which is option C!

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QuestionFind the reference angle. A) $\frac{11 \pi}{36}$ B) $\frac{\pi}{9}$ C) $\frac{5 \pi}{18}$ D) $\frac{2 \pi}{9}$

What is this asking? We need to find the smallest angle between the terminal side of $\frac{41\pi}{18}$ and the x-axis. Watch out! Reference angles are always positive and less than or equal to $\frac{\pi}{2}$ .

1. Revolutions
2. Find the coterminal angle
3. Determine the quadrant and calculate the reference angle

Let's rewrite our angle $\frac{41\pi}{18}$ in terms of full revolutions ( $2\pi$ ).
We can think of $\frac{41\pi}{18}$ as a mixed number of revolutions!

We divide $\frac{41}{18}$ to get $2$ with a remainder of $5$ .
So, $\frac{41\pi}{18}$ is the same as $2\pi + \frac{5\pi}{18}$ .
This means we have two full revolutions and an extra $\frac{5\pi}{18}$ radians!

Coterminal angles are angles that share the same terminal side.
Since $\frac{41\pi}{18}$ is the same as two full revolutions plus $\frac{5\pi}{18}$ , the coterminal angle we're interested in is just $\frac{5\pi}{18}$ .

Since $\frac{5\pi}{18}$ is between $0$ and $\frac{\pi}{2}$ (which is $\frac{9\pi}{18}$ ), our angle is in the first quadrant!

For angles in the first quadrant, the reference angle is simply the angle itself.
Therefore, the reference angle is $\frac{5\pi}{18}$ .

The reference angle for $\frac{41\pi}{18}$ is $\frac{5\pi}{18}$ , which is option C!