Reference Angle For 120 Degrees

Reference Angle

In math, a reference angle is generally an acute angle enclosed between the terminal arm and the x-centrality. It is e'er positive and less than or equal to 90 degrees. Let the states learn more most the reference angle in this article.

ane.	Reference Bending Definition
2.	Rules for Reference Angles in Each Quadrant
3.	How to Find Reference Angles?
four.	FAQs on Reference Angle

Reference Bending Definition

The reference bending is the smallest possible bending fabricated by the terminal side of the given angle with the x-axis. It is e'er an acute angle (except when it is exactly ninety degrees). A reference bending is always positive irrespective of which side of the axis it is falling.

How to Draw Reference Bending?

To draw the reference bending for an angle, identify its final side and see by what bending the terminal side is close to the 10-axis. The reference bending of 135° is drawn below:

reference angle

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

Rules for Reference Angles in Each Quadrant

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

Quadrant	Bending, θ	Reference Angle Formula in Degrees	Reference Bending Formula in Radians
I	lies betwixt 0° and 90°	θ	θ
II	lies between 90° and 180°	180 - θ	π - θ
3	lies between 180° and 270°	θ - 180	θ - π
IV	lies betwixt 270° and 360°	360 - θ	2π - θ

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

how to find reference angle

Instance: Find the reference angle of 120°.

Solution: The given angle is, θ = 120°. We know that 120° lies in quadrant 2. Using the higher up rules, its reference bending is,

180 - θ = 180 - 120 = 60°

Therefore, the reference bending of 120° is 60°.

How to Find Reference Angles?

In the previous section, we learned that nosotros could detect the reference angles using the set of rules mentioned in the table. That table works only when the given angle lies between 0° and 360°. But what if the given angle does not lie in this range? Permit's run into how we can find the reference angles when the given angle is greater than 360°.

Steps to Discover Reference Angles

The steps to find the reference angle of an bending are explained with an case. Allow us discover the reference bending of 480°.

Stride 1: Find the coterminal angle of the given angle that lies betwixt 0° and 360°.

The coterminal angle can exist institute either past calculation or subtracting 360° from the given angle as many times equally required. Let'southward find the coterminal angle of 480° that lies between 0° and 360°. Nosotros will subtract 360° from 480° to observe its coterminal bending.

480° - 360° = 120°

Step two: If the angle from footstep one lies betwixt 0° and 90°, and so that angle itself is the reference bending of the given angle. If not, and so nosotros have to check whether it is closest to 180° or 360° and by how much.

Here, 120° does not lie between 0° and 90° and information technology is closest to 180° past 60°. i.due east.,

180° - 120° = 60°

Step 3: The angle from step 2 is the reference bending of the given bending.

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

This is how we can find reference angles of any given bending.

► Important Notes:

The reference bending of an angle is always non-negative i.e., a negative reference angle doesn't be.
The reference angle of any angle always lies betwixt 0 and π/2 (both inclusive).

Tricks to Find Reference Angles:

We use the reference angle to find the values of trigonometric functions at an angle that is beyond xc°. For example, nosotros tin can see 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 because 135° is in quadrant II, where sine is positive.

sin 495° = √2/2 [Using unit circle]

If we use reference angles, we don't demand to remember the complete unit circle, instead we tin simply remember the commencement quadrant values of the unit circumvolve.

Reference Bending Examples

Example 1: Notice the reference angle of 8π/3 in radians.

Solution: The given angle is greater than 2π.

Step ane: Finding co-final angle:

We find its co-terminal bending by subtracting 2π from information technology.

8π/iii - 2π = 2π/3

This angle does not lie between 0 and π/2. Hence, it is not the reference angle of the given angle.

Step ii: Finding reference angle:

Allow's cheque whether 2π/iii is close to π or 2π and by how much. Conspicuously, 2π/3 is close to π by π - 2π/3 = π/3. Therefore, the reference angle of 8π/3 is π/three.
Example ii: Using the above example (Example i), find cos (8π/3).

Solution: In the above case, nosotros found that the coterminal angle of 8π/3 is 2π/three which lies in quadrant 2 (as it lies between π/two and π), where cos is negative.

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

= - cos (π/3)

= -1/2 (Using the trigonometry table)

Therefore, cos (8π/3) = -i/2.
Case three: What is the reference angle of -1200°?

Solution: The given angle is -1200°.

Stride one: Finding the co-terminal angle:

We run into that 1200/360 = 3.333...

This indicates that there are three complete angles (360° angles) in 1200°. To detect a positive co-final angle of -1200°, we need to add four multiples of 360° to it. (If we add merely 3 multiples of 360° to it, the reply won't be a positive angle).

-1200° + 4(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 close to 180° and past sixty°. Therefore, the reference angle of -1200° is 60°.

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Practice Questions on Reference Bending

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FAQs on Reference Angle

What is a Reference Angle?

A reference angle is an bending bounded between the terminal arm and the x-centrality. Information technology is a positive acute bending lies between 0° to 90° or a xc degree bending. It is of import to sympathize the reference angle as information technology has its applications in finding the values of trigonometric ratios and in representing trigonometric functions on graphs.

How do you Find the Reference Angle?

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

The get-go step is to find the coterminal angle of the given angle that lies between 0° to 360°. It is done past adding or subtracting 360° or 2π from the given angle as many times as required. So, in the case of 500°, if we decrease 360° from it, we will become 500° - 360° = 140°.
The next step is to check whether the angle obtained in pace 1 (140°) is closer to 180° or 360° and by how much. Here, 140° is closer to 180° by 40°.
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?

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

Can Reference Angles be Negative?

A reference angle is a not-negative angle. Information technology is always positive and cannot be negative in measurement.

How to Find Reference Bending in Radians?

To notice reference angles in radians is the aforementioned as finding them in degrees. The merely difference is that in radians we supplant 180° past π and 360° by 2π. Follow the rules given below to find reference angles in radians:

Quadrant one - θ
Quadrant two - π - θ
Quadrant 3 - θ - π
Quadrant four - 2π - θ

How to Find Reference Bending of Negative Angle?

To find the reference angle of a negative angle, we have to add 360° or 2π to it as many times every bit required to observe its coterminal angle. For case, to find the reference angle of -1000°, nosotros volition add together 360° three times to it. It implies, - 1000° + iii(360°) = -1000° + 1080° = 80°. Therefore, fourscore° is the required reference angle of a negative angle of -1000°. If θ in a negative angle -θ is from 0 to 90 degrees, then its reference angle is θ. For case, the reference angle of -78° is 78°.

What is the Reference Angle for 7π/half-dozen?

The adding to find the reference angle of 7π/6 is given beneath:

7π/vi lies in the 3rd quadrant, so,

Reference angle = 7π/6 - π

= π/6

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

How to Observe Reference Bending in Quadrant 3?

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