To find the coterminal angle and reference angle for the given angles θ, you can follow these steps:
(a) θ = 15π/4:
Coterminal Angle:
To find coterminal angles, you can add or subtract multiples of 2π (one full revolution) from the given angle. In this case:
Coterminal Angle = 15π/4 + 2πn, where n is an integer.
You can choose different values of n to find different coterminal angles.
Reference Angle:
The reference angle for θ is the positive acute angle between the terminal side of θ and the x-axis. In this case, θ = 15π/4 is already in standard position (it's on the positive x-axis), so the reference angle is simply 0.
(b) θ = -17π/5:
Coterminal Angle:
Similarly, for this angle:
Coterminal Angle = -17π/5 + 2πn, where n is an integer.
Again, you can choose different values of n to find different coterminal angles.
Reference Angle:
To find the reference angle, consider the positive acute angle between the terminal side of θ and the x-axis. Since θ is negative and points in the opposite direction, you can find its positive counterpart by taking the absolute value:
Positive θ = |-17π/5| = 17π/5
Now, the reference angle is the acute angle formed by the positive θ:
Reference Angle = π - (17π/5 - π)
Reference Angle = π - 17π/5 + π
Reference Angle = 2π - 17π/5
Now, simplify the reference angle:
Reference Angle = (10π/5) - (17π/5)
Reference Angle = (10π - 17π)/5
Reference Angle = (-7π)/5
So, for θ = -17π/5, the coterminal angles are given by -17π/5 + 2πn, and the reference angle is (-7π)/5.