数学学習

三角方程式 | 定義、例、解き方

例とステップバイステップの方法で三角方程式を解く

Interactive Trigonometric Equations Visualization

Welcome to the Trigonometric Equations Explorer

This interactive visualization helps you understand how to solve trigonometric equations by finding where the function intersects with a target value. Explore different types of equations and see how periodicity creates multiple solutions.

What you can explore:

  • Basic sine equations - sin(x) = k with two solutions per period
  • Basic cosine equations - cos(x) = k with two solutions per period
  • Basic tangent equations - tan(x) = k with one solution per period
  • Squared equations - sin²(x) and cos²(x) with four solutions per period
  • Linear combinations - More complex equations with transformations

How to Use This Visualization

Interactive Features:

  • • Select Equation Type - Choose from 6 different trigonometric equations
  • • Adjust Periods - Control how many periods to display
  • • Toggle Visualizations - Show/hide solutions, target line, and intersections
  • • View Solutions - See all solutions in the visible domain

What You'll See:

  • • Function Curve (colored) - The trigonometric function
  • • Target Line (red, dashed) - The value we're solving for
  • • Solution Points (green) - Where function equals target value
  • • Vertical Lines (green, dashed) - Connect solutions to x-axis
  • • Solution List - All solutions with exact and decimal values

Basic sine equation with two solutions per period

More periods show more solutions

Solutions

x1

0.524

≈ 0.524

x2

2.618

≈ 2.618

x3

6.807

≈ 6.807

x4

8.901

≈ 8.901

x5

13.090

≈ 13.090

x6

15.184

≈ 15.184

Total: 6 solutions

Equation Info

Equation: sin(x) = 0.5

Target: 0.5

Periods: 4

Solving Strategy

Step 1: Isolate the trigonometric function (sin, cos, or tan).

Step 2: Find the reference angle using inverse trigonometric functions.

Step 3: Determine all solutions in one period using symmetry.

Step 4: Add period multiples to find all solutions: x = base + n·period.

Key Concepts

Periodicity: sin and cos have period 2π, tan has period π. Solutions repeat every period.

Multiple Solutions: Most trigonometric equations have infinitely many solutions due to periodicity.

General Solution: Express solutions as x = base + n·period where n is any integer.

General Solution Formulas

For sin(x) = k:

x = arcsin(k) + 2nπ

x = π - arcsin(k) + 2nπ

For cos(x) = k:

x = arccos(k) + 2nπ

x = -arccos(k) + 2nπ

For tan(x) = k:

x = arctan(k) + nπ

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