Wize High School Algebra II Textbook (Common Core) > Trigonometric Ratios (Radians)

Solving Simple Trigonometric Equations

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Solving Simple Trigonometric Equations
A simple trigonometric equation is in the form:
f(θ)=kf(\theta)=kf(θ)=k

where k∈Rk\in\mathbb{R}k∈R and f(θ)f(\theta)f(θ) is a trigonometric function. 

Step 1. 
Identify what quadrant(s) the angle lies in. 

Step 2.
Identify the reference angle. 

Step 3. 
Solve for the solutions in the appropriate quadrants. 

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Example 1

Solve for θ\thetaθ over the domain 0≤θ<2π0\leq{\theta}<2\pi0≤θ<2π.

cos⁡θ=12\cos{\theta}=\displaystyle\frac{1}{2}cosθ=21​

Step 1. 
Identify what quadrant(s) the angle lies in. 

Since cos⁡θ>0\cos{\theta}>0cosθ>0, then cos⁡θ\cos{\theta}cosθ is in quadrant I & IV. 

Step 2.
Identify the reference angle. 

Since cos⁡θ=x\cos{\theta}=xcosθ=x, then we need the angle that corresponds to an x-coordinate point of 12\displaystyle\frac{1}{2}21​.
The reference angle is π3\displaystyle\frac{\pi}{3}3π​.

Step 3. 
Solve for the solutions in the appropriate quadrants. 

Quadrant I: 

π3\displaystyle\frac{\pi}{3}3π​

Quadrant IV: 

2π−π3=5π32\pi-\displaystyle\frac{\pi}{3}=\displaystyle\frac{5\pi}{3}2π−3π​=35π​

The solutions are θ=π3, 5π3\theta=\displaystyle\frac{\pi}{3},~\displaystyle\frac{5\pi}{3}θ=3π​, 35π​

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Example: Solving Simple Trigonometric Equations

Solve cos⁡θ=−32\cos{\theta}=\displaystyle-\frac{\sqrt{3}}{2}cosθ=−23​​ :

a. Over the domain 0≤θ<360∘0\leq{}\theta<360^\circ0≤θ<360∘

b. Provide the general solution

a.

Step 1. 

Since cos⁡θ<0,\cos{\theta}<0,cosθ<0, then θ\thetaθ is in quadrant II & III.

Step 2.

The reference angle is π6\displaystyle\frac{\pi}{6}6π​

Step 3. 

Quadrant II: π−π6=5π6\pi-\displaystyle\frac{\pi}{6}=\displaystyle\frac{5\pi}{6}π−6π​=65π​

Quadrant III; π+π6=7π6\pi+\displaystyle\frac{\pi}{6}=\displaystyle\frac{7\pi}{6}π+6π​=67π​

b. 
The general solution is all coterminal angles to the solutions in Step 3. 

Therefore, 

5π6±2nπ,  n∈W\displaystyle\frac{5\pi}{6}\pm2n\pi,~~n\in\mathbb{W}65π​±2nπ,  n∈W

7π6±2nπ,  n∈W\displaystyle\frac{7\pi}{6}\pm2n\pi,~~n\in\mathbb{W}67π​±2nπ,  n∈W

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Example: Solving Simple Trigonometric Equations

Solve csc⁡θ=−2\csc{\theta}=-\sqrt{2}cscθ=−2​ :

a. Over the domain 0≤θ<360°0\le\theta<360^{\degree }0≤θ<360°

b. Provide the general solution

a.

Step 1. 

Since csc⁡θ<0,\csc{\theta}<0,cscθ<0, then θ\thetaθ is in quadrant III & IV.

Step 2.

The reference angle is π4\displaystyle\frac{\pi}{4}4π​

Step 3. 

Quadrant II: π+π4=5π4\pi+\displaystyle\frac{\pi}{4}=\displaystyle\frac{5\pi}{4}π+4π​=45π​

Quadrant III; 2π−π4=11π42\pi-\displaystyle\frac{\pi}{4}=\displaystyle\frac{11\pi}{4}2π−4π​=411π​

b. 
The general solution is all coterminal angles to the solutions in Step 3. 

Therefore, 

5π4±2nπ,  n∈W\displaystyle\frac{5\pi}{4}\pm2n\pi,~~n\in\mathbb{W}45π​±2nπ,  n∈W

11π4±2nπ,  n∈W\displaystyle\frac{11\pi}{4}\pm2n\pi,~~n\in\mathbb{W}411π​±2nπ,  n∈W

Practice: Solving Simple Trigonometric Equations
Solve cot⁡θ=0\cot{\theta}=0cotθ=0 over the domain 0≤θ<2π0\leq{}\theta<2\pi0≤θ<2π.

A) 3π2\displaystyle\frac{3\pi}{2}23π​ 

B) π2\displaystyle\frac{\pi}{2}2π​ 

C) π2, 3π2\displaystyle\frac{\pi}{2},~\displaystyle\frac{3\pi}{2}2π​, 23π​ 

D) 0, π0,~\pi0, π 

I don't know

Practice: Solving Simple Trigonometric Equations

Let 3cos⁡θ+23cos⁡θ=27\sqrt{3}\cos{\theta}+2\sqrt{3}\cos{\theta}=\sqrt{27}3​cosθ+23​cosθ=27​. Solve for θ \theta~θ  over the domain 0≤θ<2π0\leq{\theta}<2\pi0≤θ<2π.

A) 000  

B) 0,π,2π0, \pi, 2\pi0,π,2π  

C) π2,3π2\displaystyle\frac{\pi}{2}, \displaystyle\frac{3\pi}{2}2π​,23π​  

D) 0,π0, \pi0,π   

I don't know

Practice: Solving Simple Trigonometric Equations
Let sin⁡2θ=12\sin{2\theta}=\displaystyle\frac{1}{2}sin2θ=21​. Solve for θ\thetaθ, providing the general solution. 

A) θ=π12,5π12±nπ    n∈Z\theta=\displaystyle\frac{\pi}{12},\displaystyle\frac{5\pi}{12}\pm{}n\pi{}~~~~n\in\mathbb{Z}θ=12π​,125π​±nπ    n∈Z 

B) θ=π6,5π6±2nπ    n∈Z\theta=\displaystyle\frac{\pi}{6},\displaystyle\frac{5\pi}{6}\pm{}2n\pi{}~~~~n\in\mathbb{Z}θ=6π​,65π​±2nπ    n∈Z 

C) θ=π12,5π12±2nπ    n∈Z\theta=\displaystyle\frac{\pi}{12},\displaystyle\frac{5\pi}{12}\pm{}2n\pi{}~~~~n\in\mathbb{Z}θ=12π​,125π​±2nπ    n∈Z 

D) θ=π3,5π3±nπ    n∈Z\theta=\displaystyle\frac{\pi}{3},\displaystyle\frac{5\pi}{3}\pm{}n\pi{}~~~~n\in\mathbb{Z}θ=3π​,35π​±nπ    n∈Z 

I don't know

Extra Practice

Solving Trigonometric Equations

Find aaa over the domain 0≤a<2π0\leq{}a<2\pi0≤a<2π
cot⁡a2=1\cot\dfrac{a}{2}=1cot2a​=1

Solving Simple Trigonometric Ratios

Solve for θ\thetaθ over the interval (−∞,∞)(-\infin,\infin)(−∞,∞):
4cos⁡3θ=24\cos3\theta=24cos3θ=2

Solving Simple Trigonometric Equations

Solve for θ\thetaθ over the domain 0≤θ<2π0\leq{}\theta<2\pi0≤θ<2π
2sin⁡θ=−22\sin\theta=-\sqrt{2}2sinθ=−2​

Solving Simple Trigonometric Equations

Practice: Solving Simple Trigonometric Equations
Let tan⁡2θ=1\tan{2\theta}=1tan2θ=1. Solve for θ\thetaθ, providing the general solution. 

Solving Simple Trigonometric Equations

Practice: Solving Simple Trigonometric Equations
Let cos⁡2θ=−32\cos{2\theta}=-\dfrac{\sqrt{3}}{2}cos2θ=−23​​. Solve for θ\thetaθ, providing the general solution.  

Solving Simple Trigonometric Equations

Practice: Solving Simple Trigonometric Equations
Let 32sec⁡θ−2sec⁡θ=43\sqrt{2}\sec{\theta}-\sqrt{2}\sec{\theta}=432​secθ−2​secθ=4. Solve for θ \theta~θ  over the domain 0≤θ<2π0\leq{\theta}<2\pi0≤θ<2π.

Solving Simple Trigonometric Equations

Practice: Solving Simple Trigonometric Equations
Let 4sin⁡θ−2sin⁡θ=14\sin\theta-2\sin\theta=14sinθ−2sinθ=1. Solve for θ \theta~θ  over the domain 0≤θ<2π0\leq{\theta}<2\pi0≤θ<2π.

Solving Simple Trigonometric Equations

Practice: Solving Simple Trigonometric Equations
Solve cos⁡θ=12\cos{\theta}=\dfrac{1}{\sqrt{2}}cosθ=2​1​ over the domain 0≤θ<2π0\leq{}\theta<2\pi0≤θ<2π.

Solving Simple Trigonometric Equations

Practice: Solving Simple Trigonometric Equations
Solve sin⁡θ=−32\sin{\theta}=-\dfrac{\sqrt{3}}{2}sinθ=−23​​ over the domain 0≤θ<2π0\leq{}\theta<2\pi0≤θ<2π.