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Trigonometric Equations and Inequalities

Alice White

Alice White

6 min read

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Study Guide Overview

This study guide covers inverse trigonometric functions, focusing on their role as the "undo" operation for trigonometric functions. It explains solving trigonometric equations and inequalities, emphasizing the importance of domain and range restrictions. It also highlights the periodic nature of trigonometric functions and provides examples and practice questions. Key concepts include arcsin, arccos, arctan, and understanding solution intervals.

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. Her...

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