Question

b. given \\(\tan\theta = \frac{8}{9}\\), which statement is true for all possible values of \\(\theta\\)?
a. \\(\cot\theta = \frac{9}{8}\\)
b. \\(\cot\theta = -\frac{9}{8}\\)
c. \\(\cot\theta = -\frac{8}{9}\\)
d. \\(\cot\theta\\) cannot be determined

1. what is the length of the arc that subtends a central angle of \\(150^\circ\\) in the unit circle?

a. \\(\frac{5}{6}\pi\\) units
b. \\(75\pi\\) units
c. \\(\frac{6}{5}\pi\\) units
d. \\(\frac{5}{12}\pi\\) units

1. in the unit circle, the length of the arc that subtends a positive central angle of \\(\theta\\) is \\(\frac{1}{3}\pi\\) units. what is the measure of \\(\theta\\) in degrees?

a. \\(120^\circ\\)
b. \\(60\pi^\circ\\)
c. \\(\frac{1}{540}^\circ\\)
d. \\(60^\circ\\)

1. in a circle with radius 8 units, the length of the arc that subtends a positive central angle of \\(\theta\\) is \\(\frac{20}{9}\pi\\) units. what is the measure of \\(\theta\\) in degrees?

a. \\(100^\circ\\)
b. \\(50\pi^\circ\\)
c. \\(50^\circ\\)
d. \\(400^\circ\\)

Explanation:

Question 8

Step1: Recall cotangent definition

Cotangent is the reciprocal of tangent, so \(\cot\theta=\frac{1}{\tan\theta}\).

Step2: Substitute \(\tan\theta = \frac{8}{9}\)

\(\cot\theta=\frac{1}{\frac{8}{9}}=\frac{9}{8}\).

Step1: Arc length formula for unit circle

Arc length \(s = r\theta\) (in radians), unit circle has \(r = 1\), so \(s=\theta\) (when \(\theta\) is in radians). First convert \(150^\circ\) to radians: \(\theta=150^\circ\times\frac{\pi}{180^\circ}=\frac{5\pi}{6}\) radians.

Step2: Calculate arc length

Since \(r = 1\), \(s=\theta=\frac{5\pi}{6}\) units.

Step1: Arc length formula for unit circle

\(s = r\theta\), \(r = 1\), so \(s=\theta\) (radians). Given \(s=\frac{1}{3}\pi\), so \(\theta=\frac{1}{3}\pi\) radians.

Step2: Convert radians to degrees

\(\theta=\frac{1}{3}\pi\times\frac{180^\circ}{\pi}=60^\circ\).

Answer:

A. \(\cot\theta=\frac{9}{8}\)

Question 9