AP Pre-Calculus: Solving Trig Equations & Inequalities 🚀

Hey there! Let's make sure you're totally ready to rock the AP Pre-Calculus exam. This guide is designed to be your go-to resource, especially the night before the test. We'll break down trig equations and inequalities, making sure everything clicks. Let's dive in!

1. Inverse Trigonometric Functions: The "Undo" Button 🔄

Think of inverse trig functions as the "undo" button for regular trig functions (sine, cosine, tangent). They help you find the angle when you know the ratio. For example:

If sin(x) = 0.5, then x = arcsin(0.5) = 30°
If cos(x) = 0.5, then x = arccos(0.5) = 60°
If tan(x) = 1, then x = arctan(1) = 45°

Key Concept

Remember, inverse trig functions give you the angle that corresponds to a specific trig ratio. They are essential for solving equations and inequalities involving trigonometric functions.

2. Solving Trigonometric Inequalities 📐

Solving inequalities is similar to solving equations, but with a twist. For example:

Equation: sin(x) = 0.5 → x = 30°
Inequality: sin(x) > 0.5 → 30° < x < 150°

Use the unit circle or your calculator to find the angles. Always check if you need to be in degree or radian mode!

Exam Tip

When solving inequalities, sketch the graph of the trig function to visualize the intervals where the inequality holds true. This will help avoid common mistakes related to interval notation.

3. Solving Trigonometric Equations 🧮

3.1 Domain Restrictions 🚧

Inverse trig functions have domain restrictions. This means they only output values within a specific range. Here’s a quick look:

arcsin(x): Outputs angles between -90° and 90°
arccos(x): Outputs angles between 0° and 180°
arctan(x): Outputs angles between -90° and 90°

Common Mistake

Forgetting domain restrictions is a common mistake. Always double-check that your solutions fall within the valid range for the inverse function you're using.

3.2 Periodic Nature of Trig Functions 🔄

Sine, cosine, and tangent are periodic. This means they repeat their values every 2π radians (or 360°). So, there are infinite solutions to trig equations. However, we often focus on solutions within a specific interval.

Memory Aid

Think of trig functions as waves – they go up and down, repeating their pattern. This repetition is why there are multiple solutions to trig equations.

3.3 Example: Solving cos(x) = -0.2

Let's solve cos(x) = -0.2:

Take the arccosine of both sides: x = arccos(-0.2)
Calculator gives: x ≈ 101.5°
Since cosine is negative in the second and third quadrants, the other solution in 0° to 360° is 360° - 101.5° = 258.5°
However, arccosine's range is 0° to 180°. So, we need to choose the solution within this range. Therefore, x ≈ 101.5°

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Image courtesy of Wiktionary.

Quick Fact

When using arccosine, the solution will always be in the first or second quadrant, i.e. between 0° and 180°.

4. Domain and Range Restrictions: A Quick Guide 🧭

Here's a handy table summarizing the domain and range restrictions for trig functions and their inverses:

Function	Domain	Range
Sine	All real numbers	[-1, 1]
Cosine	All real numbers	[-1, 1]
Tangent	All real numbers, except π/2 + kπ	All real numbers
Arcsine	[-1, 1]	[-π/2, π/2]
Arccosine	[-1, 1]	[0, π]
Arctangent	All real numbers	(-π/2, π/2)

Understanding domain and range restrictions is crucial for solving trig equations and inequalities correctly. Pay close attention to these details, as they are often tested on the AP exam.

Final Exam Focus 🎯

High-Priority Topics: Inverse trig functions, solving equations and inequalities, domain and range restrictions.
Common Question Types: Multiple-choice questions testing your understanding of domain and range, free-response questions requiring you to solve trig equations and inequalities within a given interval.
Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
Common Pitfalls: Forgetting domain restrictions, using the wrong mode on your calculator, not considering the periodic nature of trig functions.

Exam Tip

Always double-check your answers, especially when dealing with inequalities. Make sure your solutions make sense in the context of the problem.

Practice Questions

Practice Question

Multiple Choice Questions

1. Which of the following is the correct solution to the equation sin(x) = 0.6, when restricted to the domain of 0 to 360 degrees?

A) 30 degrees B) 150 degrees C) 210 degrees D) 330 degrees

Answer: B) 150 degrees

2. Which of the following is the correct solution to the inequality cos(x) > 0.5, when restricted to the domain of 0 to 360 degrees?

A) x < 30 degrees or x > 150 degrees B) x > 30 degrees and x < 150 degrees C) x < 30 degrees or x > 210 degrees D) x > 30 degrees and x < 210 degrees

Answer: B) x > 30 degrees and x < 150 degrees

3. Which of the following is the correct solution to the equation tan(x) = -2, when restricted to the domain of -90 to 90 degrees?

A) -63.4 degrees B) -116.6 degrees C) 116.6 degrees D) None of the above

Answer: A) -63.4 degrees

Free Response Question

Question: Consider the function f(x) = 2sin(x) - 1. (a) Find all solutions to the equation 2sin(x) - 1 = 0 in the interval [0, 2π]. (3 points) (b) Solve the inequality 2sin(x) - 1 > 0 in the interval [0, 2π]. (4 points) (c) Sketch the graph of f(x) = 2sin(x) - 1 in the interval [0, 2π] and label all x-intercepts. (3 points)

Answer Key:

(a)

2sin(x) - 1 = 0
2sin(x) = 1
sin(x) = 1/2 (1 point)
x = π/6, 5π/6 (2 points - 1 point each)

(b)

2sin(x) - 1 > 0
sin(x) > 1/2 (1 point)
π/6 < x < 5π/6 (3 points)

(c) Graph should show a sine wave with amplitude 2, shifted down by 1, and x-intercepts at π/6 and 5π/6. (3 points)

You've got this! Remember to stay calm, take deep breaths, and trust in your preparation. You're going to do great! 💪

Trigonometric Equations and Inequalities

Alice White