What Happens With A Negative Coterminal Angle How Do You Find The Reference Angle

Reference Bending

In math, a reference bending is generally an astute angle enclosed between the concluding arm and the ten-centrality. It is always positive and less than or equal to 90 degrees. Let us learn more about the reference angle in this article.

i.	Reference Angle Definition
two.	Rules for Reference Angles in Each Quadrant
iii.	How to Discover Reference Angles?
4.	FAQs on Reference Angle

Reference Angle Definition

The reference angle is the smallest possible bending fabricated by the terminal side of the given angle with the ten-centrality. It is ever an astute angle (except when it is exactly 90 degrees). A reference angle is always positive irrespective of which side of the axis it is falling.

How to Describe Reference Bending?

To draw the reference angle for an angle, identify its terminal side and see by what angle the terminal side is close to the x-axis. The reference bending of 135° is fatigued beneath:

reference angle

Here, 45° is the reference angle of 135°.

Rules for Reference Angles in Each Quadrant

Hither are the reference angle formulas depending on the quadrant of the given angle.

Quadrant	Bending, θ	Reference Bending Formula in Degrees	Reference Angle Formula in Radians
I	lies between 0° and xc°	θ	θ
II	lies between xc° and 180°	180 - θ	π - θ
Three	lies between 180° and 270°	θ - 180	θ - π
4	lies between 270° and 360°	360 - θ	2π - θ

If the angle is in radians, then we use the same rules as for degrees by replacing 180° with π and 360° with 2π.

how to find reference angle

Instance: Detect the reference angle of 120°.

Solution: The given angle is, θ = 120°. We know that 120° lies in quadrant Ii. Using the in a higher place rules, its reference angle is,

180 - θ = 180 - 120 = threescore°

Therefore, the reference angle of 120° is sixty°.

How to Find Reference Angles?

In the previous section, we learned that nosotros could find the reference angles using the set up of rules mentioned in the table. That tabular array works but when the given angle lies betwixt 0° and 360°. But what if the given angle does not lie in this range? Let'due south meet how we can find the reference angles when the given angle is greater than 360°.

Steps to Find Reference Angles

The steps to find the reference angle of an bending are explained with an example. Allow united states detect the reference bending of 480°.

Step 1: Observe the coterminal angle of the given angle that lies between 0° and 360°.

The coterminal angle can exist constitute either by adding or subtracting 360° from the given angle as many times as required. Let's detect the coterminal angle of 480° that lies between 0° and 360°. We will subtract 360° from 480° to find its coterminal angle.

480° - 360° = 120°

Step 2: If the angle from footstep 1 lies between 0° and 90°, then that angle itself is the reference angle of the given bending. If not, and so we accept to bank check whether it is closest to 180° or 360° and past how much.

Hither, 120° does not lie between 0° and 90° and it is closest to 180° past lx°. i.eastward.,

180° - 120° = lx°

Step three: The angle from step 2 is the reference angle of the given bending.

Thus, the reference angle of 480° is 60°.

This is how we can detect reference angles of any given angle.

► Important Notes:

The reference angle of an angle is always non-negative i.due east., a negative reference angle doesn't exist.
The reference bending of whatever angle always lies between 0 and π/2 (both inclusive).

Tricks to Discover Reference Angles:

We utilize the reference angle to find the values of trigonometric functions at an bending that is beyond xc°. For example, we tin encounter that the coterminal angle and reference angle of 495° are 135° and 45° respectively.

sin 495° = sin 135° = +sin 45°.

We have included the + sign considering 135° is in quadrant II, where sine is positive.

sin 495° = √ii/2 [Using unit circle]

If we employ reference angles, we don't need to remember the complete unit circle, instead we can merely remember the start quadrant values of the unit circle.

Reference Bending Examples

Instance 1: Find the reference angle of 8π/3 in radians.

Solution: The given angle is greater than 2π.

Footstep 1: Finding co-terminal angle:

We find its co-final angle past subtracting 2π from it.

8π/3 - 2π = 2π/3

This bending does not prevarication betwixt 0 and π/2. Hence, it is not the reference angle of the given angle.

Step 2: Finding reference angle:

Let's check whether 2π/3 is close to π or 2π and by how much. Clearly, 2π/3 is close to π by π - 2π/3 = π/3. Therefore, the reference bending of 8π/3 is π/iii.
Example two: Using the in a higher place example (Example 1), find cos (8π/3).

Solution: In the above example, we plant that the coterminal angle of 8π/3 is 2π/3 which lies in quadrant Ii (as it lies betwixt π/two and π), where cos is negative.

Also, we found that the reference angle of 8π/3 is π/3. And then, cos (8π/3) = - cos (π/3).

= - cos (π/3)

= -1/two (Using the trigonometry tabular array)

Therefore, cos (8π/iii) = -1/2.
Example 3: What is the reference angle of -1200°?

Solution: The given bending is -1200°.

Step 1: Finding the co-terminal angle:

We encounter that 1200/360 = 3.333...

This indicates that at that place are iii complete angles (360° angles) in 1200°. To find a positive co-terminal angle of -1200°, nosotros need to add 4 multiples of 360° to information technology. (If we add together just three multiples of 360° to it, the reply won't be a positive bending).

-1200° + iv(360°) = 240°

Step two: Finding the reference angle:

Let's check whether 240° is close to 180° or 360° and by how much. Clearly, 240°is shut to 180° and past sixty°. Therefore, the reference angle of -1200° is sixty°.

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Practice Questions on Reference Angle

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FAQs on Reference Bending

What is a Reference Angle?

A reference angle is an angle divisional betwixt the terminal arm and the 10-axis. It is a positive acute angle lies between 0° to xc° or a 90 caste angle. Information technology is important to sympathize the reference bending as it has its applications in finding the values of trigonometric ratios and in representing trigonometric functions on graphs.

How do you Find the Reference Bending?

To observe the reference angle. let's say of 500°, follow the steps given below:

The first stride is to find the coterminal angle of the given angle that lies between 0° to 360°. Information technology is done by adding or subtracting 360° or 2π from the given angle as many times as required. So, in the case of 500°, if nosotros decrease 360° from it, we will get 500° - 360° = 140°.
The next step is to cheque whether the angle obtained in footstep ane (140°) is closer to 180° or 360° and past how much. Here, 140° is closer to 180° past twoscore°.
This angle is the reference angle of the given angle. Therefore, 40° is the reference angle of 500°.

What is the Reference Angle for a 200° Angle?

Betwixt the angles 180° and 360°, we tin can say that 200° is close to 180° by xx°. Thus, the reference angle of 200° is twenty°.

Tin can Reference Angles exist Negative?

A reference bending is a non-negative bending. It is always positive and cannot be negative in measurement.

How to Observe Reference Angle in Radians?

To find reference angles in radians is the same as finding them in degrees. The only difference is that in radians nosotros supervene upon 180° past π and 360° by 2π. Follow the rules given below to find reference angles in radians:

Quadrant 1 - θ
Quadrant 2 - π - θ
Quadrant 3 - θ - π
Quadrant 4 - 2π - θ

How to Find Reference Bending of Negative Angle?

To find the reference bending of a negative angle, we have to add 360° or 2π to it equally many times as required to find its coterminal angle. For instance, to discover the reference angle of -1000°, we will add 360° three times to it. It implies, - k° + 3(360°) = -chiliad° + 1080° = fourscore°. Therefore, 80° is the required reference bending of a negative angle of -chiliad°. If θ in a negative angle -θ is from 0 to xc degrees, then its reference angle is θ. For example, the reference bending of -78° is 78°.

What is the Reference Angle for 7π/6?

The calculation to notice the reference angle of 7π/6 is given beneath:

7π/6 lies in the third quadrant, so,

Reference bending = 7π/6 - π

= π/six

Therefore, the reference angle for 7π/6 is π/6.

How to Find Reference Bending in Quadrant 3?

If an bending θ is given which lies in the 3rd quadrant, then its reference bending can be found by using the formula θ - π.

Source: https://www.cuemath.com/geometry/reference-angle/

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