Chapter
Chapter 7
Section
7.6
Solutions 52 Videos

Factor each expression.

\sin^2\theta - \sin\theta

0.27mins
Q1a

Factor each expression.

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

0.25mins
Q1b

Factor each expression.

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

0.32mins
Q1c

Factor each expression.

4\cos^2\theta - 1

0.22mins
Q1d

Factor each expression.

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

0.53mins
Q1e

Factor each expression.

49 \tan^2{x} - 64

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}}

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

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

1.04mins
Q2c

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.

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

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.

2.40mins
Q3

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

\sin^2\theta = 1

0.26mins
Q4a

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

0.28mins
Q4b

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

1.00mins
Q4c

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

1.09mins
Q4d

Solve each equation for x, where 0^{\circ} \leq x \leq 360^{\circ}.

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

0.40mins
Q5a

Solve each equation for x, where 0^{\circ} \leq x \leq 360^{\circ}.

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

0.45mins
Q5b

Solve each equation for x, where 0^{\circ} \leq x \leq 360^{\circ}.

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

0.38mins
Q5c

Solve each equation for x, where 0^{\circ} \leq x \leq 360^{\circ}.

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

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

1.06mins
Q5e

Solve each equation for x, where 0^{\circ} \leq x \leq 360^{\circ}.

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

0.34mins
Q5f

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

1.03mins
Q6a

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

0.17mins
Q6b

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

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

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

1.32mins
Q6e

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

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

1.15mins
Q7a

Solve for \theta to the nearest hundredth, where 0 \leq \theta \leq 2\pi.

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

1.19mins
Q7b

Solve for \theta to the nearest hundredth, where 0 \leq \theta \leq 2\pi.

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

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

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

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

2.43mins
Q7f

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

1.12mins
Q8a

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

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

2.00mins
Q8c

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

2.48mins
Q8d

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

1.42mins
Q8e

Solve each equation for x, where 0 \leq x \leq 2\pi.

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

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

2.08mins
Q9a

Solve each equation in the interval 0 \leq x \leq 2\pi. Round to two decimal places, if necessary.

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 

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 

2.12mins
Q9d

Solve the equation 8\sin^2x - 8\sin x + 1= 0 in the interval 0\leq x \leq 2\pi

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?

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 Natasha’s home, less than zero?

3.39mins
Q13

Solve the equation 6\sin^2x = 17\cos x + 11 for x in the interval 0\leq x \leq 2\pi

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).

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).

2.01mins
Q17

Solve the equation 2\cos 3x + \cos 2x + 1 = 0 in the interval 0\leq x\leq 2\pi

6.42mins
Q18

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

3.11mins
Q19

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

4.04mins
Q20
Lectures 11 Videos

ex. Solve for x when 0 \leq x \leq 2\pi (Quadratic Equation for Trig)

\sin^2x -\sin x = 2

2.53mins

ex. Solve for x when 0 \leq x \leq 2\pi (Quadratic Equation for Trig)

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

1.27mins

ex. Solve for x when 0 \leq x \leq 2\pi (Quadratic Equation for Trig)

2\sin^2x -1 = 0

2.28mins

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

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

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

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

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

3.28mins

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}

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