How do I find the reference angle in radicals given the value of cosecant 5 pi/6?
If the given angle is θ, and the reference angle is Ø, then Ø is always in QI.
If θ is in Q1, Ø = θ
in QII, Ø = π-θ
in QIII, Ø = θ-π
in QIV, Ø = 2π-θ
so, in this case, your reference angle is pi - 5pi/6 = pi/6
To find the reference angle of a given angle in radians, you need to follow a few steps. In this case, we are given the value of cosecant (csc) of an angle, which is equal to 1/sine (sin) of the angle.
Step 1: Determine the sine function
To find the reference angle, we need to identify the value of the sine function. We know that csc(θ) = 1/sin(θ), so let's start by finding the value of sin(θ).
Step 2: Use the reciprocal identity
Since csc(θ) = 1/sin(θ), we can rearrange the equation to solve for sin(θ), yielding sin(θ) = 1/csc(θ).
Step 3: Convert to sine function
We now have sin(θ) = 1/csc(θ). Plug in the given value of csc(5π/6) to find the value of sin(5π/6) = 1/csc(5π/6).
Step 4: Evaluate the sine function
To find sin(5π/6), we need to recall the unit circle. The sine function at 5π/6 (or 150°) is positive because the y-coordinate of the corresponding unit circle point is positive. In the case of 5π/6, the value of sin(5π/6) = 1/2.
Step 5: Find the reference angle
The reference angle is the acute angle that is formed between the terminal side of the given angle (5π/6 in this case) and the x-axis. Since 5π/6 lies in the second quadrant, the reference angle would be the difference between 5π/6 and π radians (180 degrees).
Reference angle = 5π/6 - π = 5π/6 - 6π/6 = -π/6 (or -30°)
Note: The reference angle is defined as a positive angle, formed between the given angle and the x-axis.
In conclusion, the reference angle for the angle 5π/6 (or 150°) is -π/6 (or -30°).