7.6 Solving Quadratic Trigonometric Equations
Chapter
Chapter 7
Section
7.6
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Lectures 11 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
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

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}

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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 Natasha’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