Math / Trigonometry

QuestionFind the reference angle. $575^{\circ}$ A) $5^{\circ}$ B) $35^{\circ}$ C) $70^{\circ}$ D) $55^{\circ}$

Studdy Solution

STEP 1

What is this asking? We need to find the smallest positive angle that $575^\circ$ is away from the x-axis. Watch out! Reference angles are always positive and between $0^\circ$ and $90^\circ$ , so if you get something outside of this range, double check your work!

STEP 2

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

STEP 3

Let's think about how many full circles, or revolutions, are in $575^\circ$ .
Since one full circle is $360^\circ$ , we can divide $575^\circ$ by $360^\circ$ to see how many full revolutions we have.

STEP 4

$\frac{575}{360} = 1 \text{ with a remainder}$ We see that $575^\circ$ contains one full revolution of $360^\circ$ .
Let's find out how much is left over.

STEP 5

To find the angle that's left after one full revolution, we subtract the full revolution ( $360^\circ$ ) from our original angle ( $575^\circ$ ).
This will give us a coterminal angle, which shares the same terminal side as our original angle but is less than a full circle.

STEP 6

$575^\circ - 360^\circ = 215^\circ$ So, $215^\circ$ is coterminal with $575^\circ$ .
They land at the same spot, but $215^\circ$ gets there without going full circle.

STEP 7

Now, let's figure out which quadrant $215^\circ$ is in.
Since $215^\circ$ is between $180^\circ$ and $270^\circ$ , it's in the third quadrant.

STEP 8

In the third quadrant, the reference angle is the angle between our coterminal angle and the x-axis, specifically the negative x-axis which is at $180^\circ$ .
To find the reference angle, we subtract $180^\circ$ from our coterminal angle: $215^\circ - 180^\circ = 35^\circ$

STEP 9

The reference angle for $575^\circ$ is $\boxed{35^\circ}$ .
So the answer is B!

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QuestionFind the reference angle. $575^{\circ}$ A) $5^{\circ}$ B) $35^{\circ}$ C) $70^{\circ}$ D) $55^{\circ}$

Studdy Solution

STEP 1

What is this asking? We need to find the smallest positive angle that $575^\circ$ is away from the x-axis. Watch out! Reference angles are always positive and between $0^\circ$ and $90^\circ$ , so if you get something outside of this range, double check your work!

STEP 2

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

STEP 3

Let's think about how many full circles, or revolutions, are in $575^\circ$ .
Since one full circle is $360^\circ$ , we can divide $575^\circ$ by $360^\circ$ to see how many full revolutions we have.

STEP 4

$\frac{575}{360} = 1 \text{ with a remainder}$ We see that $575^\circ$ contains one full revolution of $360^\circ$ .
Let's find out how much is left over.

STEP 5

To find the angle that's left after one full revolution, we subtract the full revolution ( $360^\circ$ ) from our original angle ( $575^\circ$ ).
This will give us a coterminal angle, which shares the same terminal side as our original angle but is less than a full circle.

STEP 6

$575^\circ - 360^\circ = 215^\circ$ So, $215^\circ$ is coterminal with $575^\circ$ .
They land at the same spot, but $215^\circ$ gets there without going full circle.

STEP 7

Now, let's figure out which quadrant $215^\circ$ is in.
Since $215^\circ$ is between $180^\circ$ and $270^\circ$ , it's in the third quadrant.

STEP 8

In the third quadrant, the reference angle is the angle between our coterminal angle and the x-axis, specifically the negative x-axis which is at $180^\circ$ .
To find the reference angle, we subtract $180^\circ$ from our coterminal angle: $215^\circ - 180^\circ = 35^\circ$

STEP 9

The reference angle for $575^\circ$ is $\boxed{35^\circ}$ .
So the answer is B!

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QuestionFind the reference angle. 575∘575^{\circ}575∘ A) 5∘5^{\circ}5∘ B) 35∘35^{\circ}35∘ C) 70∘70^{\circ}70∘ D) 55∘55^{\circ}55∘

Studdy Solution

STEP 1

What is this asking? We need to find the smallest positive angle that 575∘575^\circ575∘ is away from the x-axis. Watch out! Reference angles are always positive and between 0∘0^\circ0∘ and 90∘90^\circ90∘, so if you get something outside of this range, double check your work!

STEP 2

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

STEP 3

Let's think about how many full circles, or revolutions, are in 575∘575^\circ575∘.Since one full circle is 360∘360^\circ360∘, we can divide 575∘575^\circ575∘ by 360∘360^\circ360∘ to see how many full revolutions we have.

STEP 4

575360=1 with a remainder \frac{575}{360} = 1 \text{ with a remainder} 360575​=1 with a remainder We see that 575∘575^\circ575∘ contains one full revolution of 360∘360^\circ360∘.Let's find out how much is left over.

STEP 5

To find the angle that's left after one full revolution, we subtract the full revolution (360∘360^\circ360∘) from our original angle (575∘575^\circ575∘).This will give us a *coterminal angle*, which shares the same terminal side as our original angle but is less than a full circle.

STEP 6

575∘−360∘=215∘ 575^\circ - 360^\circ = 215^\circ 575∘−360∘=215∘ So, 215∘215^\circ215∘ is *coterminal* with 575∘575^\circ575∘.They land at the same spot, but 215∘215^\circ215∘ gets there without going full circle.

STEP 7

Now, let's figure out which quadrant 215∘215^\circ215∘ is in.Since 215∘215^\circ215∘ is between 180∘180^\circ180∘ and 270∘270^\circ270∘, it's in the **third quadrant**.

STEP 8

STEP 9

The reference angle for 575∘575^\circ575∘ is 35∘\boxed{35^\circ}35∘​.So the answer is B!

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Start learning now

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QuestionFind the reference angle. $575^{\circ}$ A) $5^{\circ}$ B) $35^{\circ}$ C) $70^{\circ}$ D) $55^{\circ}$

What is this asking? We need to find the smallest positive angle that $575^\circ$ is away from the x-axis. Watch out! Reference angles are always positive and between $0^\circ$ and $90^\circ$ , so if you get something outside of this range, double check your work!

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

Let's think about how many full circles, or revolutions, are in $575^\circ$ .
Since one full circle is $360^\circ$ , we can divide $575^\circ$ by $360^\circ$ to see how many full revolutions we have.

$\frac{575}{360} = 1 \text{ with a remainder}$ We see that $575^\circ$ contains one full revolution of $360^\circ$ .
Let's find out how much is left over.

To find the angle that's left after one full revolution, we subtract the full revolution ( $360^\circ$ ) from our original angle ( $575^\circ$ ).
This will give us a coterminal angle, which shares the same terminal side as our original angle but is less than a full circle.

$575^\circ - 360^\circ = 215^\circ$ So, $215^\circ$ is coterminal with $575^\circ$ .
They land at the same spot, but $215^\circ$ gets there without going full circle.

Now, let's figure out which quadrant $215^\circ$ is in.
Since $215^\circ$ is between $180^\circ$ and $270^\circ$ , it's in the third quadrant.

The reference angle for $575^\circ$ is $\boxed{35^\circ}$ .
So the answer is B!