Understanding Trigonometric Equations through Key Angles
Trigonometry is a fundamental branch of mathematics with wide-ranging applications in various fields, from physics to engineering. One of the key aspects of trigonometry involves solving trigonometric equations, especially those involving key angles like 45°. Let's explore a series of problems and their solutions, emphasizing the importance of these key angles and the application of basic trigonometric ratios.
1. Solving Equations with Trigonometric Ratios
Given that sin 45° 1/√2, consider the equation:
2 sin x -√2
To solve this, we can rewrite the given equation as:
2 sin x -2/√2
Dividing both sides by 2, we get:
sin x -1/√2
From trigonometric values, we know that:
sin(-45°) -1/√2
Thus, the solution is:
x -45° 360°n, where n is an integer.
2. Applying Trigonometric Identities and Square Roots
Let's consider another equation:
sin x √2 -sin x
Rearranging and squaring both sides, we get:
(sin x √2 sin x)^2 0
This simplifies to:
(2√2 sin x)^2 0
Further simplification leads to:
2√2 sin x 0
Dividing by 2√2, we find:
sin x 0
The primary solutions within one period are:
x 0° and 180°
3. Solving with sin x √2 sin x
Consider another scenario:
sin x √2 sin x
This can be rewritten as:
2 sin x √2
Dividing both sides by 2, we get:
sin x 1/√2
From trigonometric values, we know:
sin 45° 1/√2
Thus, the general solution is:
x 45° 360°n, where n is an integer.
4. Another Approach with Key Angles
Let's explore another equation:
sin x √2 -sin x
This can be rewritten as:
2 sin x -√2
Thus,
sin x -1/√2
The primary solution is:
x 225° and 315°
In radians:
x 5π/4 and 7π/4
5. Applying Trigonometric Ratios and Key Angles
Let's solve this equation:
sin x 2^1/2 -sin x
This can be rewritten as:
2 sin x -√2
Thus,
sin x -1/√2
From trigonometric values, we know that:
sin 225° -1/√2
Thus, the general solution is:
x 225° 360°n and 315° 360°n, where n is an integer.
Conclusion
Trigonometric equations involving key angles like 45°, 90°, and 180° are essential for solving various problems. Familiarity with basic trigonometric ratios and identities simplifies the process significantly. The solutions to these equations and the application of key angles can be crucial for solving complex problems in mathematics and related fields.