Textbook

Advanced Functions McGraw-Hill
Chapter

Chapter 5
Section

Cumulative Review of Trig Ch4 and 5

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Solutions
22 Videos

a) Determine the exact radian measure of `100^o`

b) Determine the exact degree measure of `\frac{7\pi}{12}`

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Q1

A gymnasium has a circular running track around the mezzanine with a radius of 20 m. A runner ran along the arc of the track for 60 m. What is the sector angle, in radians, from the start to the finish of his run?

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Q2

Determine an exact value for `\sin \frac{3\pi}{4} - \tan \frac{5\pi}{4}`

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Q3

Two guy wires are attached to the top of a radio antenna 25 m in height. The wires make
angles of ```
\displaystyle
\frac{\pi}{6}
```

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

with the ground, as shown.

Determine an exact expression for the distance between the two anchor points of the wires, A and B.

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Q4

Given that `\cos \frac{7\pi}{15} \doteq 0.1045`

, use an equivalent expression to determine `\sin \frac{29\pi}{30}`

, to four decimal places.

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Q5

If `\sin y = \cos 3y`

, determine an exact valeu for `\angle y`

.

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Q6

Prove ```
\displaystyle
\frac{\sin 2x}{\sec x} = \frac{2\cos^2x}{\csc x}
```

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Q10

Prove ```
\displaystyle
\sin(x + y)\cos(x -y) = \frac{\sin x}{\sec x} + \frac{\cos y}{\csc y}
```

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Q11

Commercial bottling machines often use a circular drum as part of a mechanism to install tops on bottles.

One such machine has a diameter of 120 cm, and makes a complete turn once every 5 s.

A sensor at the left side of the drum monitors its movement. Take the sensor position as zero.

a) Model the horizontal position of a point on the drum, `h`

, in centimetres, as a function of time, `t`

, in seconds.

b) Sketch a graph of h versus t over two cycles.

c) If the technician monitoring the machine increases the speed to complete a cycle in 3 5, what changes would occur in your model? Justify your answer.

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Q17

Find `x \in [0, 2\pi]`

to the nearest hundredth of a radian.

```
\displaystyle
\sec x - 5 =0
```

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Q18a

Find `x \in [0, 2\pi]`

to the nearest hundredth of a radian.

```
\displaystyle
12 \sin^2x - \sin x - 1= 0
```

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Q18b

As a science project, Anwar monitored the content of carbon monoxide outside his house in the city over several days. He found that it reached a maximum of about 30 ppm (parts per million) at 6:00 P.M., and a minimum of 10 ppm at 6:00 A.M.

a) Select a point on the graph where the instantaneous rate of change of the carbon monoxide level appears to be a maximum.

b) Use a method similar to that in Example 1 of Section 5.5 to estimate the instantaneous rate of change of the carbon monoxide level at this point, to one decimal place.

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Q21