Question

multiple choice: (1 mark each)

1. an angle of $\frac{16\pi}{3}$, expressed in degrees, is:

a) $16.8^{\circ}$ b) $-120^{\circ}$ c) $960^{\circ}$ d) $240^{\circ}$

1. if $\tan\theta=\frac{a}{b}$ and $\sin\theta = -\frac{a}{3}$, then $\sec\theta$ is equal to:

a) $\frac{3}{a}$ b) $-\frac{3}{a}$ c) $\frac{3}{b}$ d) $-\frac{3}{b}$

1. if $\cos\theta=-\frac{40}{41}$ and $\pi<\theta<\frac{3\pi}{2}$, then $\sin\theta$ is equal to:

a) $\frac{9}{41}$ b) $-\frac{9}{41}$ c) $\frac{1}{41}$ d) $-\frac{1}{41}$

1. the coordinates of the point, on the unit circle, on the terminal arm of an angle of $2640^{\circ}$ are:

a) $\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\
ight)$ b) $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\
ight)$ c) $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\
ight)$ d) $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\
ight)$

1. the terminal arm of angle $\theta$, in standard position, passes through the point $(7, - 3)$. the exact value of $\cos\theta$ can be expressed in the form of $\frac{a\sqrt{b}}{c}$. the value of $a$ is:

a) $7$ b) $3$ c) $-3$ d) $-7$

Explanation:

Step1: Convert radians to degrees

We know that to convert radians to degrees, we use the formula \( \text{Degrees}=\frac{180^{\circ}}{\pi}\times\text{Radians}\).
For \( \theta = \frac{16\pi}{3}\), we have \( \text{Degrees}=\frac{180^{\circ}}{\pi}\times\frac{16\pi}{3}\).
Cancel out the \( \pi\) terms: \( \frac{180\times16}{3}\).
Calculate \( 180\div3 = 60\), then \( 60\times16=960^{\circ}\).

Step2: Analyze the range of angles

Since \( 960^{\circ}=2\times360^{\circ}+240^{\circ}\), but when we just do the conversion \( \frac{180^{\circ}}{\pi}\times\frac{16\pi}{3}=960^{\circ}\) directly (the formula \( \theta_{degrees}=\frac{180}{\pi}\theta_{radians}\) gives the equivalent degree - measure without considering coterminal angles in a reduced - range sense for the multiple - choice context here).

Answer:

C. \(960^{\circ}\)