Solving Trigonometric Inequalities Practice Test

Test Objectives

Demonstrate the ability to solve a trigonometric equation
Demonstrate the ability to solve a trigonometric inequality

Solving Trigonometric Inequalities Practice Test:

#1:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}1 + \text{sin}\hspace{.1em}x > \frac{2 - \sqrt{2}}{2}$$

$$b)\hspace{.1em}2\hspace{.1em}\text{tan}\hspace{.1em}x < -2\sqrt{3}$$

Watch the Step by Step Video Solution | View the Written Solution

#2:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}2\text{cos}\left(2x - \frac{π}{6}\right) ≤ 1$$

$$b)\hspace{.1em}{-}3 + \frac{2}{5}\text{sin}\left(2x - \frac{5π}{6}\right) < -\frac{16}{5}$$

Watch the Step by Step Video Solution | View the Written Solution

#3:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}1 - \text{cos}\hspace{.1em}x > \text{sin}\hspace{.1em}x$$

$$b)\hspace{.1em}{-}\text{csc}\hspace{.1em}x + \text{cot}\hspace{.1em}x ≥ -1$$

Watch the Step by Step Video Solution | View the Written Solution

#4:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}\text{cos}\hspace{.1em}2x + 2 > -3\text{cos}\hspace{.1em}x$$

$$b)\hspace{.1em}\text{sin}^2 2x ≥ 3\text{cos}^2 x$$ Note: The period for y = 3cos²x is π and the period for y = sin²2x is π/2

Watch the Step by Step Video Solution | View the Written Solution

#5:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}{-}2\text{csc}\hspace{.1em}x \hspace{.1em}\text{sin}\hspace{.1em}x ≤ -\sqrt{2}\text{csc}\hspace{.1em}x$$

$$b)\hspace{.1em}\text{cos}\hspace{.1em}x + 1 ≥ -\sqrt{3}\text{cos}\hspace{.1em}\frac{x}{2}$$

Watch the Step by Step Video Solution | View the Written Solution

Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\frac{7π}{4}+ 2πn < x < \frac{13π}{4}+ 2πn$$ Graph of y=sin(x) + sqrt(2)/2

$$b)\hspace{.1em}\frac{π}{2}+ πn < x < \frac{2π}{3}+ πn$$ Graph of y=tan(x) + sqrt(3)

Watch the Step by Step Video Solution

#2:

Solutions:

$$a)\hspace{.1em}\frac{π}{4}+ πn ≤ x ≤ \frac{11π}{12}+ πn$$ Graph of y=2cos(2x - pi/6) - 1

$$b)\hspace{.1em}πn < x < \frac{π}{3}+ πn$$ Graph of y=2sin(2x - 5pi/6) + 1

Watch the Step by Step Video Solution

#3:

Solutions:

$$a)\hspace{.1em}\frac{π}{2}+ 2πn < x < 2π + 2πn$$ Graph of y=1 - cos(x) - sin(x)

$$b)\hspace{.1em}2πn < x ≤ \frac{π}{2}+ 2πn$$ $$\text{or}$$ $$π + 2πn < x < 2π + 2πn$$ Graph of y=-csc(x) + cot(x) + 1 Note: There are holes in the graph at 2πn, where n is any integer. If you are using an online graphing calculator, this may not be obvious as they are using an estimation algorithm to sketch the graph. This comes from the fact that -csc(x) + cot(x) + 1 gets really close to 1 as x approaches 2πn. We can't actually plug in 2πn because csc(x) and cot(x) are undefined. There are vertical asymptotes at π + 2πn, where n is any integer. As x approaches π + 2πn, -csc(x) + cot(x) + 1 approaches either positive or negative infinity.

Watch the Step by Step Video Solution

#4:

Solutions:

$$a)\hspace{.1em}\frac{4π}{3}+ 2πn < x < \frac{8π}{3}+ 2πn$$ Graph of y=cos(2x) + 3cos(x) + 2

$$b)\hspace{.1em}\frac{π}{3}+ πn ≤ x ≤ \frac{2π}{3}+ πn$$ Graph of y=sin^2(2x) - 3cos^2(x)

Watch the Step by Step Video Solution

#5:

Solutions:

$$a)\hspace{.1em}\frac{π}{4}+ 2πn ≤ x ≤ \frac{3π}{4}+ 2πn$$ $$\text{or}$$ $$π + 2πn < x < 2π + 2πn$$ Graph of y=-2csc(x)sin(x) + sqrt(2)csc(x)

$$b)\hspace{.1em}\frac{5π}{3}+ 4πn ≤ x ≤ \frac{7π}{3}+ 4πn$$ $$\text{or}$$ $$3π + 4πn ≤ x ≤ 5π + 4πn$$ Graph of y=cos(x) + sqrt(3)cos(x/2) + 1

Watch the Step by Step Video Solution