Textbook

Advanced Functions Nelson
Chapter

Chapter 7
Section

Chapter Test - Trig Identities and Equations

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Solutions
17 Videos

Prove that ```
\displaystyle
\frac{1-2\sin^2x}{\cos x + \sin x} + 2\sin \frac{x}{2}\cos \frac{x}{2} = \cos x
```

.

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Q1

Solve for x. ```
\displaystyle
\cos 2x + 2\sin^2x - =-2
```

, where `0\leq x \leq 2\pi`

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Q2

Determine the solution(s) for the of the following equation, where `0\leq x \leq 2\pi`

.

```
\displaystyle
\cos x = \frac{\sqrt{3}}{2}
```

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Q3a

Determine the solution(s) for the of the following equation, where `0\leq x \leq 2\pi`

.

```
\displaystyle
\tan x = - \sqrt{3}
```

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Q3b

Determine the solution(s) for the of the following equation, where `0\leq x \leq 2\pi`

.

```
\displaystyle
\sin x = - \frac{\sqrt{2}}{2}
```

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Q3c

The quadratic trigonometric equation `a \cos2 x +b\cos x - 1 = 0`

has the solutions `\frac{\pi}{3}, \pi`

, and `\frac{5\pi}{3}`

in the interval `0 \leq x \leq 2\pi`

. What are the values of `a`

and `b`

?

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Q4

The depth of the ocean at a swim buoy can be modelled by the
function `d(t) = 4 + 2\sin(\frac{\pi}{6}t)`

, where `d`

is the depth of water in metres and t is the time in hours, if `0 \leq t \leq 24`

. Consider a day when `t = 0`

represents midnight. Determine when the depth of water is 3 m.

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Q5

Nina needs to find the cosine of `\frac{11\pi}{4}`

. If she knows the sine and cosine of `\pi`

, as well as the sine and cosine of `\frac{7\pi}{4}`

, how can she find the cosine of `\frac{11\pi}{4}`

? What is her answer?

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Q6

Solve `3 \sin x + 2 = 1.5`

, where `0\leq x \leq 2\pi`

.

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Q7

The tangent of the acute angle a is `0.75`

, and the tangent of the acute angle `\beta`

is `2.4`

. Without using a calculator, determine the value of
`\sin (\alpha - \beta)`

and `\cos(\alpha + \beta)`

.

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Q8

The angle `x`

lies in the interval `\frac{\pi}{2} \leq x \leq \pi`

, and `\sin^2x = \frac{4}{9}`

. Determine the value of the following. State your answer in exact values.

```
\displaystyle
\sin 2x
```

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Q9a

The angle `x`

lies in the interval `\frac{\pi}{2} \leq x \leq \pi`

, and `\sin^2x = \frac{4}{9}`

. Determine the value of the following. State your answer in exact values.

```
\displaystyle
\cos 2x
```

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Q9b

The angle `x`

lies in the interval `\frac{\pi}{2} \leq x \leq \pi`

, and `\sin^2x = \frac{4}{9}`

. Determine the value of the following. State your answer in exact values.

```
\displaystyle
\cos \frac{x}{2}
```

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Q9c

`x`

lies in the interval `\frac{\pi}{2} \leq x \leq \pi`

, and `\sin^2x = \frac{4}{9}`

. Determine the value of the following. State your answer in exact values.

```
\displaystyle
\sin 3x
```

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Q9d

Solve for `x`

when `-2\pi \leq x \leq 2\pi`

.

```
\displaystyle
2-14\cos x = -5
```

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Q10a

Solve for `x`

when `-2\pi \leq x \leq 2\pi`

.

```
\displaystyle
9-22\cos x - 1= 19
```

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Q10b

Solve for `x`

when `-2\pi \leq x \leq 2\pi`

.

```
\displaystyle
2 + 7.5 \cos x = -5.5
```

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Q10c