What is the reference angle for (27 pi)/5?
There are 2 pi radians in the units circle, so you want to take (27/5)pi % 2pi. '%' means modulus, or to take the remainder after dividing.
See "Maths" by "Annalee" a few topics down for more details.
Forgot the second step.
Once you find (27/5)pi % 2pi = 1.4pi, to find the reference angle look on the units circle to the closest horizontal axis.
1.4pi is closer to 1*pi than 0 rad or 2pi rad. Because 1.4pi is greater than pi, your reference angle is 1.4pi - pi = 0.4pi.
If your angle was less than 1*pi, but greater than pi/2 (so it is still closer to pi then 0), you would need to subtract that angle from pi.
For example, if your reference angle is 0.8pi, 0.8pi is still closer to pi then 0. So your reference angle is pi - 0.8pi = 0.2pi
To find the reference angle for (27π/5), follow these steps:
Step 1: Calculate the equivalent positive angle within one full revolution.
- Since 2π is one complete revolution, we can find the equivalent angle by dividing 27π by 2π: (27π/5) / (2π) = 27/10.
- So, the equivalent positive angle within one full revolution is 27/10.
Step 2: Determine if the angle is in the first or second quadrant.
- Since (27π/5) is greater than π/2 (which represents a right angle), the angle is in the second quadrant.
Step 3: Find the complementary angle in the second quadrant.
- The complementary angle in the second quadrant is equal to (π - θ), where θ is the angle. In this case, θ = 27/10.
- So, the complementary angle in the second quadrant is π - (27/10) = (10π/10) - (27/10) = (10π - 27)/10.
Therefore, the reference angle for (27π/5) is (10π - 27)/10.
To find the reference angle for a given angle, we need to understand that the reference angle is always the acute angle formed between the terminal side of the angle and the x-axis.
In this case, the given angle is (27 pi)/5. To determine its reference angle, we can follow these steps:
Step 1: Convert the given angle to its equivalent angle within one complete revolution (2π radians).
- To do this, we divide the angle by 2π: (27 π)/5 ÷ 2π = 27/10.
- This means that (27 π)/5 is equivalent to 2π * (27/10), which simplifies to (54 π)/5.
Step 2: Subtract 2π (one full revolution) from the equivalent angle obtained in Step 1 until the resulting angle is between 0 and 2π (0 and 360 degrees).
- (54 π)/5 - 2π = (54 π)/5 - (10 π)/5 = (44 π)/5.
- This subtraction is done because we are looking for the angle within one full revolution.
Step 3: Since the resulting angle obtained in Step 2 is still greater than 2π, we repeat Step 2 until the angle is within one full revolution.
- (44 π)/5 - 2π = (44 π)/5 - (10 π)/5 = (34 π)/5.
Step 4: Again, check if the angle is within one full revolution.
- (34 π)/5 - 2π = (34 π)/5 - (10 π)/5 = (24 π)/5.
Now, the resulting angle, (24 π)/5, is within one full revolution (0 to 2π). The reference angle for (27 π)/5 is the acute angle formed by (24 π)/5 and the x-axis.
Therefore, the reference angle for (27 π)/5 is (24 π)/5.