Trigonometric Equations and Inequalities

This video tutorial covers AP precalculus topic 310, focusing on trigonometric equations and inequalities. It begins with an introduction to the uniqueness of trigonometric equations, followed by methods for solving them using algebra and the unit circle. The video explains how to find solutions for trigonometric equations in different ranges, including principal and all values. It also covers approaches to solve cosine and tangent equations, using factoring and the zero product rule. Finally, the tutorial addresses solving inequalities involving trigonometric functions.

What makes trigonometric functions unique compared to other types of equations?

• They have a finite number of solutions.

• They can only be solved graphically.

• They involve complex numbers.

• They are periodic and have infinite solutions.

When solving the equation 2sin(x) + 1 = 0, what is the first algebraic step?

• Multiply both sides by 2.

• Subtract 1 from both sides.

• Add 1 to both sides.

• Divide both sides by 2.

What is the principal value range for sine on the unit circle?

• 0 to π

• -π to π

• -π/2 to π/2

• 0 to 2π

How do you find all solutions for a trigonometric equation using the unit circle?

• Find the principal value and add 2πk.

• Find the principal value and subtract πk.

• Find the principal value and divide by k.

• Find the principal value and multiply by k.

What is the period of the tangent function?

• 4π

• π/2

• π

• 2π

When solving 2cos(x) = 1, what is the next step after dividing by 2?

• Find the inverse cosine.

• Subtract 1.

• Add 1.

• Multiply by 2.

What is the solution for cos(x) = -2?

• x = 5π/3

• x = π/3

• No solution

• x = 2π

How do you solve the inequality 2sin(x) + 5 < 4?

• Subtract 5 and divide by 2.

• Add 5 and multiply by 2.

• Subtract 4 and divide by 2.

• Add 4 and multiply by 2.

What is the range of x for which sin(x) > 1/2 on the unit circle?

• 0 to π/2

• π/3 to 2π/3

• π/6 to 5π/6

• π/4 to 3π/4

What is the correct approach to solve trigonometric inequalities graphically?

• Use a pie chart.

• Use the unit circle.

• Use a bar graph.

• Use a number line.