Solving Trigonometric Equations

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Questions and Answers

What are equations containing at least one trigonometric function called?

Trigonometric equations

Why do trigonometric equations have an infinite number of solutions?

The periodicity of trigonometric function

What is the general solution of the equation sin x = 0?

x = nπ, where n is an integer

What is the general solution of the equation cos x = 1/2?

x = (π/3) + 2nπ or x = (5π/3) + 2nπ, where n is an integer Signup and view all the answers

What is the general solution of the equation tan x = √3?

x = (π/3) + nπ, where n is an integer Signup and view all the answers

What is the domain of the function csc x?

x ≠ nπ, where n is an integer Signup and view all the answers

What is the period of the function sin x?

2π Signup and view all the answers

Which of the following is a solution to the equation sin(x) = 1?

π/2 Signup and view all the answers

What is the range of the function sin x?

[-1, 1] Signup and view all the answers

Trigonometric equations always have a finite number of solutions.

False Signup and view all the answers

Solving trigonometric equations involves finding all possible values of the variable that satisfy the equation.

True Signup and view all the answers

Trigonometric identities can be used to simplify trigonometric equations.

True Signup and view all the answers

Squaring both sides of a trigonometric equation always results in valid solutions.

False Signup and view all the answers

The general solution of a trigonometric equation represents all possible solutions.

True Signup and view all the answers

Which of the following is a trigonometric equation?

sin(x) = cos(x) Signup and view all the answers

Which of the following is NOT a trigonometric function?

exponential Signup and view all the answers

What is the amplitude of the function 3sin(x)?

3 Signup and view all the answers

Study Notes

Trigonometric Equations

Trigonometric equations are equations containing at least one trigonometric function.
Examples include sin x = 1/2, sec x = tan x, sin²x secx
Trigonometric equations have infinite solutions due to the periodicity of trigonometric functions.
To solve trigonometric equations:
- Find the solution within one period (often 0 to 2π)
- Determine the general solution by accounting for periodic nature.

Solving Trigonometric Equations

Example 1: Solve sin x = 1/2
- Reference angle for sin x = 1/2 is π/6
- Positive sin in quadrants I and II
- Solutions in [0, 2π]: x = π/6 and 5π/6
- General solution: x = π/6 + 2nπ and 5π/6 + 2nπ, where n is an integer.
Example 2: Solve 1 + cos x = 0
- cos x = -1
- Cosine is negative in quadrants II and III
- Solution in [0, 2π]: x = π
- General solution: x = π + 2nπ, where n is an integer.
Example 3: Solve 4 cos²x - 3 = 0
- cos²x = ¾
- cos x = ±√(3/4) = ±√3/2
- Determine four quadrants angles:
  - cos x = √3/2: x = π/6, 11π/6
  - cos x = -√3/2: x = 5π/6, 7π/6
- General solution: x = π/6 + 2nπ, 5π/6 + 2nπ, 7π/6 + 2nπ, 11π/6 + 2nπ, where n is an integer.

General Solution of Trigonometric Equations

Solving equations with more than one trigonometric function often involves identities and algebraic manipulation to isolate one function.
Identify the period of the trigonometric function(s) to obtain the general solution.
Example 1: Solve sin x + cos x = 0
- tan x = -1
- x = 3π/4 and 7π/4
- General solution: x = 3π/4 + nπ, where n is an integer.
Example 2: Solve sin x cos x = √3/4
- This transforms to sin 2x = √3/2
- Reference angle is π/3
- Solutions in [0, 2π]: 2x = π/3, 2π/3
- Solutions in [0, 2π]: x = π/6, π/3, 7π/6, 4π/3
General solutions: x = π/6 + nπ, π/3 +nπ, 7π/6 +nπ and 4π/3 + nπ, where n is an integer.

Key Concepts

Identities and algebraic manipulation are crucial for simplifying equations
Extraneous solutions can arise when squaring both sides of an equation. These must be checked.
Solutions must be presented in the form that fits the interval.

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Description

This quiz covers the methods and examples for solving trigonometric equations, including the periodic nature of their solutions. You'll explore how to find solutions within one period and general solutions based on various equations like sin x = 1/2 and 1 + cos x = 0. Test your understanding of these concepts to master trigonometric equations.