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Solutions
41 Videos

Determine the approximate radian measure, to the nearest hundredth.

```
\displaystyle
33^o
```

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0.22mins

Q1a

Determine the approximate radian measure, to the nearest hundredth.

```
\displaystyle
138^o
```

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0.29mins

Q1b

Determine the approximate radian measure, to the nearest hundredth.

```
\displaystyle
252^o
```

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0.34mins

Q1c

Determine the approximate radian measure, to the nearest hundredth.

```
\displaystyle
347^o
```

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0.28mins

Q1d

Determine the approximate degree measure, to the nearest tenth, for each angle.

1.24

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0.26mins

Q2a

Determine the approximate degree measure, to the nearest tenth, for each angle.

2.82

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0.29mins

Q2b

Determine the approximate degree measure, to the nearest tenth, for each angle.

4.78

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0.15mins

Q2c

Determine the approximate degree measure, to the nearest tenth, for each angle.

6.91

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0.17mins

Q2d

Determine the exact radian measure.

```
\displaystyle
75^o
```

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0.19mins

Q3a

Determine the exact radian measure.

```
\displaystyle
20^o
```

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0.20mins

Q3b

Determine the exact radian measure.

```
\displaystyle
12^o
```

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0.21mins

Q3c

Determine the exact radian measure.

```
\displaystyle
9^o
```

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0.15mins

Q3d

Determine the exact degree measure of each angle.

```
\displaystyle
\frac{2\pi}{5}
```

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0.13mins

Q4a

Determine the exact degree measure of each angle.

```
\displaystyle
\frac{4\pi}{9}
```

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0.11mins

Q4b

Determine the exact degree measure of each angle.

```
\displaystyle
\frac{7\pi}{12}
```

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0.27mins

Q4c

Determine the exact degree measure of each angle.

```
\displaystyle
\frac{11\pi}{18}
```

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0.13mins

Q4d

The turntable in a microwave oven rotates 12 times per minute while the oven is operating. Determine the angular velocity of the turntable in

**a)** degrees per second

**b)** radians per second

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1.18mins

Q5

Turntables for playing vinyl records have four speeds, in revolutions per minute (rpm): 16, `33\frac{1}{3}, 45`

, and `78`

. Determine the angular velocity for each speed in

- degrees per second

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1.07mins

Q6a

Turntables for playing vinyl records have four speeds, in revolutions per minute (rpm): 16, `33\frac{1}{3}, 45`

, and 78. Determine the angular velocity for each speed in

- radians per second

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1.16mins

Q6b

Determine an exact value for each expression.

```
\displaystyle
\frac{\cot\frac{\pi}{4}}{\cos\frac{\pi}{3}\csc \frac{\pi}{2}}
```

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0.29mins

Q8a

Determine an exact value for each expression.

```
\displaystyle
\cos \frac{\pi}{6} \csc \frac{\pi}{3} + \sin \frac{\pi}{4}
```

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0.28mins

Q8b

A ski lodge is constructed with one side along a vertical
cliff such that it has a height of 15 m, as shown. Determine an exact measure for the base of the lodge, `b`

.

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1.06mins

Q9

Given that `\cot \frac{2\pi}{7} = \tan z`

, first express `\frac{2\pi}{7}`

as a difference between `\frac{\pi}{2}`

and an angle, and then apply a cofunction identity to determine the measure of angle `z`

.

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0.46mins

Q10

Given that `\cos \frac{5\pi}{9} = -\sin y`

, first express `\frac{5\pi}{9}`

as a sum of `\frac{\pi}{2}`

and an angle, and then apply a trigonometric identity to determine the measure of angle `y`

.

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1.00mins

Q11

Given that `\tan\frac{4\pi}{9} \doteq 5.6713`

, determine the following, to four decimal places, without using a calculator. Justify your answers.

```
\displaystyle
\cot \frac{\pi}{18}
```

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1.07mins

Q12a

Given that `\tan\frac{4\pi}{9} \doteq 5.6713`

, determine the following, to four decimal places, without using a calculator. Justify your answers.

```
\displaystyle
\tan \frac{13\pi}{9}
```

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0.33mins

Q12b

Given that `\tan \sin x = \cos \frac{3\pi}{11}`

and that `x`

lies in the second quadrant, determine the measure of angle `x`

.

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0.55mins

Q13

Use an appropriate compound angle formula to express as a single trigonometric function, and then determine an exact value of each.

```
\displaystyle
\sin \frac{5\pi}{12}\cos\frac{\pi}{4} + \cos\frac{5\pi}{12}\sin\frac{\pi}{4}
```

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0.36mins

Q14a

Use an appropriate compound angle formula to express as a single trigonometric function, and then determine an exact value of each.

```
\displaystyle
\sin\frac{5\pi}{12} \cos \frac{\pi}{4} - \cos \frac{5\pi}{12} \sin \frac{\pi}{4}
```

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0.25mins

Q14b

Use an appropriate compound angle formula to express as a single trigonometric function, and then determine an exact value of each.

```
\displaystyle
\cos \frac{5\pi}{12}\cos \frac{\pi}{4} - \sin \frac{5\pi}{12} \sin \frac{\pi}{4}
```

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0.35mins

Q14c

```
\displaystyle
\cos \frac{5\pi}{12} \cos \frac{\pi}{4} + \sin \frac{5\pi}{12} \sin \frac{\pi}{4}
```

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0.33mins

Q14d

Angles `x`

and `y`

are located in the first quadrant such that `\sin x =\frac{4}{5}`

and `\cos y = \frac{7}{25} `

.

- Determine an exact value for
`\cos x`

.

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0.22mins

Q15a

Angles `x`

and `y`

are located in the first quadrant such that `\sin x =\frac{4}{5}`

and `\cos y = \frac{7}{25} `

.

- Determine an exact value for
`\sin y`

.

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0.39mins

Q15b

Angles `x`

and `y`

are located in the first quadrant such that `\sin x =\frac{4}{5}`

and `\cos y = \frac{7}{25} `

.

- Determine an exact value for
`\sin(x + y)`

.

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0.54mins

Q15c

Angle x lies in the third quadrant, and `\tanx = \frac{7}{24}`

.

**a)** Determine an exact value of `\cos 2x`

.

**b)** Determine an exact value of `\sin 2x`

.

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2.01mins

Q16

Determine an exact value for `\cos \frac{13\pi}{12}`

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1.25mins

Q17

Prove that ```
\displaystyle
\sin(2\pi -x ) = -\sin x
```

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0.25mins

Q18

Prove that ```
\displaystyle
\sec x = \frac{2(\cos x \sin 2x - \sin x \cos 2x)}{\sin 2x}
```

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1.31mins

Q19a

Prove that ```
\displaystyle
2\sin x \cos y = \sin(x + y) + \sin(x - y)
```

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0.30mins

Q20

Consider the equation `\cos 2x = 2 \sin x \sec x`

. Either prove that it is an identity, or determine a counterexample to show that it is not an identity.

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0.30mins

Q22

Prove that ```
\displaystyle
(\sin 2x)(\tan x + \cot x) =2
```

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0.38mins

Q23