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Solutions
52 Videos

Factor each expression.

`\sin^2\theta - \sin\theta`

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0.27mins

Q1a

Factor each expression.

`\cos^2\theta - 2\cos\theta +1`

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0.25mins

Q1b

Factor each expression.

`3\sin^2\theta - \sin\theta - 2`

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0.32mins

Q1c

Factor each expression.

`4\cos^2\theta - 1`

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0.22mins

Q1d

Factor each expression.

`24\sin^2x - 2\sin{x} - 2`

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0.53mins

Q1e

Factor each expression.

`49 \tan^2{x} - 64`

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0.39mins

Q1f

Solve the first equation in each pair of equations for `y`

and/or `z`

. Then
use the same strategy to solve the second equation for `x`

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

.

`y^2 = \displaystyle{\frac{1}{3}}, \tan^2x = \displaystyle{\frac{1}{3}}`

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1.28mins

Q2a

Solve the first equation in each pair of equations for `y`

and/or `z`

. Then
use the same strategy to solve the second equation for `x`

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

.

`y^2+y = 0, \sin^2x + \sin{x} = 0`

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0.48mins

Q2b

Solve the first equation in each pair of equations for `y`

and/or `z`

. Then
use the same strategy to solve the second equation for `x`

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

.

`y-2yz = 0, \cos{x} - 2\cos{x}\sin{x} = 0`

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1.04mins

Q2c

`y`

and/or `z`

. Then
use the same strategy to solve the second equation for `x`

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

.

`yz = y, \tan{x}\sec{x} = \tan{x}`

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1.12mins

Q2d

**a)** Solve the equation `6y^2-y-1=0`

.

**b)** Solve `6\cos^2{x} - \cos{x}-1=0`

for `0 \leq x \leq 2\pi`

.

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2.40mins

Q3

Solve for `\theta`

, to the nearest degree, in the interval `0^{\circ} \leq \theta \leq 360^{\circ}`

.

`\sin^2\theta = 1`

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0.26mins

Q4a

Solve for `\theta`

, to the nearest degree, in the interval `0^{\circ} \leq \theta \leq 360^{\circ}`

.

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0.28mins

Q4b

Solve for `\theta`

, to the nearest degree, in the interval `0^{\circ} \leq \theta \leq 360^{\circ}`

.

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1.00mins

Q4c

Solve for `\theta`

, to the nearest degree, in the interval `0^{\circ} \leq \theta \leq 360^{\circ}`

.

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1.09mins

Q4d

Solve each equation for `x`

, where `0^{\circ} \leq x \leq 360^{\circ}`

.

`\sin{x}\cos{x}=0`

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0.40mins

Q5a

Solve each equation for `x`

, where `0^{\circ} \leq x \leq 360^{\circ}`

.

`\sin{x}(\cos{x}-1)=0`

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0.45mins

Q5b

Solve each equation for `x`

, where `0^{\circ} \leq x \leq 360^{\circ}`

.

`(\sin{x}+1)\cos{x}=0`

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0.38mins

Q5c

Solve each equation for `x`

, where `0^{\circ} \leq x \leq 360^{\circ}`

.

`\cos{x}(2\sin{x}-\sqrt{3})=0`

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0.59mins

Q5d

Solve each equation for `x`

, where `0^{\circ} \leq x \leq 360^{\circ}`

.

`(\sqrt{2}\sin{x}-1)(\sqrt{2}\sin{x}+1)=0`

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1.06mins

Q5e

Solve each equation for `x`

, where `0^{\circ} \leq x \leq 360^{\circ}`

.

`(\sin{x}-1)(\cos{x}+1)=0`

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0.34mins

Q5f

Solve each equation for `x`

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

.

`(2\sin{x}-1)\cos{x}=0`

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1.03mins

Q6a

Solve each equation for `x`

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

.

`(\sin{x}+1)^2=0`

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0.17mins

Q6b

Solve each equation for `x`

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

.

`(2\cos{x}+\sqrt{3})\sin{x}=0`

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1.20mins

Q6c

Solve each equation for `x`

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

.

`(2\cos{x}-1)(2\sin{x}+\sqrt{3})=0`

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1.56mins

Q6d

Solve each equation for `x`

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

.

`(\sqrt{2}\cos{x}-1)(\sqrt{2}\cos{x}+1)=0`

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1.32mins

Q6e

Solve each equation for `x`

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

.

`(\sin{x}+1)(\cos{x}-1)=0`

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0.33mins

Q6f

Solve for `\theta`

to the nearest hundredth, where `0 \leq \theta \leq 2\pi`

.

`2\cos^2\theta+\cos\theta-1=0`

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1.15mins

Q7a

Solve for `\theta`

to the nearest hundredth, where `0 \leq \theta \leq 2\pi`

.

`2\sin^2\theta=1-\sin\theta`

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1.19mins

Q7b

Solve for `\theta`

to the nearest hundredth, where `0 \leq \theta \leq 2\pi`

.

`\cos^2\theta=2+\cos\theta`

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0.53mins

Q7c

Solve for `\theta`

to the nearest hundredth, where `0 \leq \theta \leq 2\pi`

.

`2\sin^2\theta + 5\sin\theta-3=0`

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1.07mins

Q7d

Solve for `\theta`

to the nearest hundredth, where `0 \leq \theta \leq 2\pi`

.

`3\tan^2\theta-2\tan\theta=1`

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2.01mins

Q7e

Solve for `\theta`

to the nearest hundredth, where `0 \leq \theta \leq 2\pi`

.

`12\sin^2\theta+\sin\theta-6=0`

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2.43mins

Q7f

Solve each equation for `x`

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

.

`\sec{x}\csc{x}-2\csc{x}=0`

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1.12mins

Q8a

Solve each equation for `x`

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

.

`3\sec^2{x}-4=0`

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1.27mins

Q8b

Solve each equation for `x`

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

.

`2\sin{x}\sec{x}-2\sqrt{3}\sin{x}=0`

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2.00mins

Q8c

Solve each equation for `x`

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

.

`2\cot{x}+\sec^2{x}=0`

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2.48mins

Q8d

Solve each equation for `x`

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

.

`\cot{x}\csc^2x=2\cot{x}`

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1.42mins

Q8e

Solve each equation for `x`

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

.

`3\tan^3x-\tan{x}=0`

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1.26mins

Q8f

Solve each equation in the interval `0 \leq x \leq 2\pi`

. Round to
two decimal places, if necessary.

`5\cos2x-\cos{x}+3=0`

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2.08mins

Q9a

Solve each equation in the interval `0 \leq x \leq 2\pi`

. Round to
two decimal places, if necessary.

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2.21mins

Q9b

Solve each equation in the interval `0 \leq x \leq 2\pi`

. Round to
two decimal places, if necessary.

```
\displaystyle
4\cos 2x + 10\sin x - 7 = 0
```

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2.05mins

Q9c

Solve each equation in the interval `0 \leq x \leq 2\pi`

. Round to
two decimal places, if necessary.

```
\displaystyle
-2\cos 2x = 2\sin x
```

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2.12mins

Q9d

Solve the equation `8\sin^2x - 8\sin x + 1= 0`

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

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2.57mins

Q10

The quadratic trigonometric equation `\cot^2x -b\cot x +c = 0`

has the solutions `\displaystyle \frac{\pi}{6}, \frac{\pi}{4}, \frac{7\pi}{6}`

, and `\frac{5\pi}{4}`

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

. What are the values of `b`

and `c`

?

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3.32mins

Q11

Natasha is a marathon runner, and she likes to train on a `2\pi`

km stretch of rolling hills. The height, in kilometres, of the hills above
sea level, relative to her home, can be modelled by the function `h(d) = 4\cos^2d - 1`

, where `d`

is the distance travelled in kilometres. At what intervals in the stretch of rolling hills is the height above sea level, relative to Natashaâ€™s home, less than zero?

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3.39mins

Q13

Solve the equation `6\sin^2x = 17\cos x + 11`

for `x`

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

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2.03mins

Q14

**(a)** Solve the equation `\sin^2x -\sqrt{2} \cos x = \cos^2x + \sqrt{2} \cos x + 2`

for `x`

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

(b) Write a general solution for the equation in part a).

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2.51mins

Q15

Given that `f(x) = \frac{\tan x}{1 - \tan x} - \frac{\cot x}{1 - \cot x}`

, determine all the values of `a`

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

, such that `f(x) = \tan(x + a)`

.

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2.01mins

Q17

Solve the equation `2\cos 3x + \cos 2x + 1 = 0`

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

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6.42mins

Q18

Solve `3\tan^22x = 1, 0^o \leq x \leq 360^o`

.

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3.11mins

Q19

Solve `\sqrt{2} \sin \theta = \sqrt{3} - \cos \theta, 0 \leq \theta \leq 2\pi`

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4.04mins

Q20

Lectures
10 Videos

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation for Trig)

`\sin^2x -\sin x = 2`

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2.53mins

Quadratic Equation Trig ex1a

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation for Trig)

`2\sin^2x -3\sin x + 1 = 0`

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1.27mins

Quadratic Equation Trig ex1b

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation for Trig)

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

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2.28mins

Quadratic Equation ex2

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation for Trig where we have `\cos`

and `\sin`

ratio)

`1 + \sin x = \cos^2x `

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2.09mins

Trig Equation that requires Identity

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation for Trig where we have `\sec`

and `\tan`

ratio)

**useful identity** ```
\displaystyle
1 + \tan^2x = \sec^2x
```

`2 \sec^2 x - 3 + \tan x = 0`

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6.30mins

Equation requiring Trig Identity ex2a

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation with Double Angles)

**useful identity** ```
\displaystyle
\cos 2x = 1 - 2\sin^2x
```

`3 \sin x + 3 \cos 2x = 2`

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4.08mins

Equation requiring Trig Identity ex2b

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Quadratic Equation with Double Angles)

**useful identity** ```
\displaystyle
\cos 2x = \cos^2 -\sin^2x
```

` \cos^2 x - \sin^2x -\frac{1}{2} = 0`

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2.48mins

Double Angle Identity ex9

*ex.* Solve for `x`

when `0 \leq x \leq \pi`

(Advanced but popular example)

**useful identity** ```
\displaystyle
A^3 + B^3 = (A+B)(A^2 +AB + B^2)
```

` \sin^6x + \cos^6x = 1`

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3.28mins

Advanced Example

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(using Compounded Angle Formula)

**useful identity** ```
\displaystyle
\sin x \cos y + \cos x \sin y = \sin(x + y)
```

` \sin x \cos\frac{\pi}{3} +\cos x \sin\frac{\pi}{3} = \frac{1}{2}`

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2.42mins

Sum Formula in Exanded format

*ex.* Solve for `x`

when `0 \leq x \leq 2\pi`

(Single Term of Sine)

**useful identity** ```
\displaystyle
\sin x \cos y + \cos x \sin y = \sin(x + y)
```

` \sin x +\cos x =1`

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2.48mins

Converting to Single Term of Sine Ex