find the angle that is between 0 and 2pi radians that is coterminal to 13pi/6 radians. The coterminal is a(pi)/b. find the value for a and b
The Correct Answer and Explanation is:
To find an angle between 0 and 2π2\pi2π that is coterminal with 13π6\frac{13\pi}{6}613π, subtract or add multiples of 2π2\pi2π until the result lies within the interval 0<θ<2π0 < \theta < 2\pi0<θ<2π.
Step 1: Express 2π2\pi2π with denominator 6
2π=12π62\pi = \frac{12\pi}{6}2π=612π
Step 2: Subtract 2π2\pi2π from 13π6\frac{13\pi}{6}613π
13π6−12π6=π6\frac{13\pi}{6} – \frac{12\pi}{6} = \frac{\pi}{6}613π−612π=6π
Step 3: Final Answer
The angle coterminal with 13π6\frac{13\pi}{6}613π and between 000 and 2π2\pi2π is:π6\frac{\pi}{6}6π
This is in the form aπb\frac{a\pi}{b}baπ, where:
- a=1a = 1a=1
- b=6b = 6b=6
Explanation
Coterminal angles differ by full rotations around the unit circle. A full rotation equals 2π2\pi2π radians. When an angle exceeds 2π2\pi2π, subtracting 2π2\pi2π reduces it back into the principal range of 000 to 2π2\pi2π. If an angle is negative, adding 2π2\pi2π accomplishes the same.
The given angle is 13π6\frac{13\pi}{6}613π. This exceeds 2π2\pi2π, which is equivalent to 12π6\frac{12\pi}{6}612π. Subtracting 12π6\frac{12\pi}{6}612π from 13π6\frac{13\pi}{6}613π yields π6\frac{\pi}{6}6π. The result lies within the desired interval 0<θ<2π0 < \theta < 2\pi0<θ<2π.
Angles like 13π6\frac{13\pi}{6}613π and π6\frac{\pi}{6}6π are coterminal because they point to the same terminal side on the unit circle. Both produce identical trigonometric values. The difference of 2π2\pi2π indicates a single full rotation between them.
Coterminal angles are useful in trigonometry and periodic functions, allowing simplification of angle measures while retaining their sine, cosine, and tangent values. This concept also aids in solving equations and understanding wave behavior.
Therefore, the equivalent angle between 000 and 2π2\pi2π is π6\frac{\pi}{6}6π, and the values of aaa and bbb are:
- a=1a = 1a=1
- b=6b = 6b=6
This expresses the final answer in the required form aπb\frac{a\pi}{b}baπ.
