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Solutions
31 Videos

An arc 33 m long subtends a central angle of a circle with a radius of 16 m. Determine the measure of the central angle in radians.

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Q1

A circle has a radius of 75 cm and a central angle of `\frac{14\pi}{15}`

. Determine the arc length.

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Q2

Convert each of the following to exact radian measure and then evaluate to one decimal.

20°

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Q3a

Convert each of the following to exact radian measure and then evaluate to one decimal.

-50°

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Q3b

Convert each of the following to exact radian measure and then evaluate to one decimal.

160°

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Q3c

Convert each of the following to exact radian measure and then evaluate to one decimal.

420

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Q3d

Convert the following to degree measure.

```
\displaystyle
\frac{\pi}{4}
```

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Q4a

Convert the following to degree measure.

```
\displaystyle
-\frac{5\pi}{4}
```

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Q4b

Convert the following to degree measure.

```
\displaystyle
\frac{8\pi}{3}
```

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Q4c

Convert the following to degree measure.

```
\displaystyle
-\frac{2\pi}{3}
```

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Q4d

For each of the following values of `\sin \theta`

, determine the measure of `\theta`

if `\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}`

.

```
\displaystyle
\frac{1}{2}
```

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Q5a

For each of the following values of `\sin \theta`

, determine the measure of `\theta`

if `\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}`

.

```
\displaystyle
-\frac{\sqrt{3}}{2}
```

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Q5b

For each of the following values of `\sin \theta`

, determine the measure of `\theta`

if `\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}`

.

```
\displaystyle
\sin \theta = \frac{\sqrt{2}}{2}
```

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Q5c

`\sin \theta`

, determine the measure of `\theta`

if `\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}`

.

```
\displaystyle
- \frac{1}{2}
```

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Q5d

If `\cos \theta = - \frac{5}{13}`

, and `0 \leq \theta \leq 2\pi`

, determine

a) `\tan \theta`

b) `\sec \theta`

c) the possible values of \theta to the nearest tenth

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Q6

A tower that is 65 m high makes an obtuse angle with the ground. The vertical distance from the top of the tower to the ground is 59 m. What obtuse angle does the tower make with the ground, to the nearest hundredth of a radian?

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Q7

State the period of the graph of each function, in radians.

(a) `y = \sin x `

(b) `y = \cos x `

(c) `y = \tan x `

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Q8

The following graph is a sine curve. Determine the equation of the graph.

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Q9

The following graph is a sine curve. Determine the equation of the graph.

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Q10

State the transformations that have been applied to `f(x) = \cos x`

to obtain each of the following functions.

```
\displaystyle
f(x) = -19\cos x - 9
```

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Q11a

State the transformations that have been applied to `f(x) = \cos x`

to obtain each of the following functions.

```
\displaystyle
f(x) = \cos(10(x + \frac{\pi}{12}))
```

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Q11b

State the transformations that have been applied to `f(x) = \cos x`

to obtain each of the following functions.

```
\displaystyle
f(x) = \frac{10}{11}\cos(x - \frac{\pi}{9}) + 3
```

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Q11c

`f(x) = \cos x`

to obtain each of the following functions.

```
\displaystyle
f(x)= -\cos(-x + \pi)
```

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Q11d

The current, `I`

, in amperes, of an electric circuit is given by the function `I(t)= 4.5 \sin (120\pi t)`

, where `t`

is the time in seconds.

a) Draw a graph that shows one cycle.

b) What is the singular period?

c) At what value of t is the current a maximum in the first cycle?

d) When is the current a minimum in the first cycle?

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Q12

State the period of the graph of each function, in radians.

(a) `y=\csc x`

(b) `y = \sec x`

(c) `y = \cot x`

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Q13

A bumblebee is flying in a circular motion within a vertical plane, at a constant speed.

The height of the bumblebee above the ground, as a function of time, can be modelled by a sinusoidal function.

At `t = 0`

, the bumblebee is at its lowest point above the ground.

a) What does the amplitude of the sinusoidal function represent in this situation?

b) What does the period of the sinusoidal function represent in this situation?

c) What does the equation of the axis of the sinusoidal function represent in this situation?

d) If a reflection in the horizontal axis was applied to the sinusoidal function, was the sine function or the cosine function used?

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Q14

The population of a ski-resort town, as a function of the number of months into the year, can be described by a cosine function. The maximum population of the town is about 15 000 people, and the minimum population is about 500 people. At the beginning of the year, the population is at its greatest. After six months, the population reaches its lowest number of people. What is the equation of the cosine function that describes the population of this town?

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Q15

A weight is bobbing up and down on a spring attached to a ceiling. The data in the following table give the height of the weight above the floor as it bobs. Determine the sine function that models this situation.

```
\displaystyle
\begin{array}{llllllll}
&t(s) &0.0 & 0.2 & 0.4 & 0.6& 0.8 & 1.0 & 1.2& 1.4 & 1.6 & 1.8 & 2.0 & 2.2 \\
&h(t) &120 & 136 & 165 & 180& 166 & 133 & 120& 135 & 164 & 179 & 165 &133
\end{array}
```

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Q16

State two intervals in which the function

```
\displaystyle
y = 7 \sin(\frac{1}{5}x) + 2
```

has na average rate of change that is

a) zero

b) a negative value

c) a positive value

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Q17

State two points where the function

```
\displaystyle
y = \frac{1}{4}\cos(4\pi x) -3
```

has an instantaneous rate of change is

a) zero

b )z negative value

c) a positive value.

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Q18

A person’s blood pressure, P(t), in millimetres of mercury (mm Hg), is modelled by the function ```
\displaystyle
P(t) = 100 -20\cos(\frac{8\pi}{3}t)
```

a) What is the period of the function?

b) What does the value of the period mean in this situation?

c) Calculate the average rate of change in a person's blood pressure on the interval `t \in [0.2, 0.3]`

.

d) Estimate the instantaneous rate of change in a person's blood pressure at `t = 0.5`

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Q19