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In math, a reference angle is generally an acute angle enclosed between the terminal arm and the x-axis. It is always positive and less than or equal to 90 degrees. Let us learn more about the reference angle in this article.
|1.||Reference Angle Definition|
|2.||Rules for Reference Angles in Each Quadrant|
|3.||How to Find Reference Angles?|
|4.||FAQs on Reference Angle|
Reference Angle Definition
The reference angle is the smallest possible angle made by the terminal side of the given angle with the x-axis. It is always an acute 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 Draw Reference Angle?
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 angle of 135° is drawn below:
Here, 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.
Reference Angle Formula in Degrees
Reference Angle Formula in Radians
|I||lies between 0° and 90°||
|II||lies between 90° and 180°||
180 - θ
|π - θ|
|III||lies between 180° and 270°||
θ - 180
|θ - π|
|IV||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.
Example: Find the reference angle of 120°.
Solution: The given angle is, θ = 120°. We know that 120° lies in quadrant II. Using the above rules, its reference angle is,
180 - θ = 180 - 120 = 60°
Therefore, the reference angle of 120° is 60°.
How to Find Reference Angles?
In the previous section, we learned that we could find 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? Let's see 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 angle are explained with an example. Let us find the reference angle of 480°.
Step 1: Find the coterminal angle of the given angle that lies between 0° and 360°.
The coterminal angle can be found either by adding or subtracting 360° from the given angle as many times as required. Let's find 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 step 1 lies between 0° and 90°, then that angle itself is the reference angle of the given angle. If not, then we 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 it is closest to 180° by 60°. i.e.,
180° - 120° = 60°
Step 3: The angle from step 2 is the reference angle of the given angle.
Thus, the reference angle of 480° is 60°.
This is how we can find reference angles of any given angle.
► Important Notes:
- The reference angle of an angle is always non-negative i.e., a negative reference angle doesn't exist.
- The reference angle of any angle always lies between 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 90°. For example, we 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 need to remember the complete unit circle, instead we can just remember the first quadrant values of the unit circle.
Related Articles on Reference Angles
Check these interesting articles related to the concept of reference angles.
Reference Angle Examples
Example 1: Find the reference angle of 8π/3 in radians.
Solution: The given angle is greater than 2π.
Step 1: Finding co-terminal angle:
We find its co-terminal angle by subtracting 2π from it.
8π/3 - 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 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 angle of 8π/3 is π/3.
Example 2: Using the above example (Example 1), find cos (8π/3).
Solution: In the above example, we found that the coterminal angle of 8π/3 is 2π/3 which lies in quadrant II (as it lies between π/2 and π), where cos is negative.
Also, we found that the reference angle of 8π/3 is π/3. So, cos (8π/3) = - cos (π/3).
= - cos (π/3)
= -1/2 (Using the trigonometry table)
Therefore, cos (8π/3) = -1/2.
Example 3: What is the reference angle of -1200°?
Solution: The given angle is -1200°.
Step 1: Finding the co-terminal angle:
We see that 1200/360 = 3.333...
This indicates that there are 3 complete angles (360° angles) in 1200°. To find a positive co-terminal angle of -1200°, we need to add 4 multiples of 360° to it. (If we add just 3 multiples of 360° to it, the answer won't be a positive angle).
-1200° + 4(360°) = 240°
Step 2: 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 by 60°. Therefore, the reference angle of -1200° is 60°.
Practice Questions on Reference Angle
FAQs on Reference Angle
What is a Reference Angle?
A reference angle is an angle bounded between the terminal arm and the x-axis. It is a positive acute angle lies between 0° to 90° or a 90 degree angle. It is important to understand the reference angle 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 Angle?
To find the reference angle. let's say of 500°, follow the steps given below:
- The first step is to find the coterminal angle of the given angle that lies between 0° to 360°. It 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 we subtract 360° from it, we will get 500° - 360° = 140°.
- The next step is to check whether the angle obtained in step 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° by 20°. Thus, the reference angle of 200° is 20°.
Can Reference Angles be Negative?
A reference angle is a non-negative angle. It is always positive and cannot be negative in measurement.
How to Find 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 we replace 180° by 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 Angle of Negative Angle?
To find the reference angle of a negative angle, we have to add 360° or 2π to it as many times as required to find its coterminal angle. For example, to find the reference angle of -1000°, we will add 360° three times to it. It implies, - 1000° + 3(360°) = -1000° + 1080° = 80°. Therefore, 80° 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 example, the reference angle of -78° is 78°.
What is the Reference Angle for 7π/6?
The calculation to find the reference angle of 7π/6 is given below:
7π/6 lies in the third quadrant, so,
Reference angle = 7π/6 - π
Therefore, the reference angle for 7π/6 is π/6.
How to Find Reference Angle in Quadrant 3?
If an angle θ is given which lies in the third quadrant, then its reference angle can be found by using the formula θ - π.