Summary of "ECUACIONES TRIGONOMETRICAS DE GRADO UNO DECIMO M2 Semana26"

Summary of the Video: "ECUACIONES TRIGONOMETRICAS DE GRADO UNO DECIMO M2 Semana26"

This video lesson focuses on solving first-degree Trigonometric Equations, emphasizing understanding, applying identities, and handling periodicity in solutions. The class includes motivation, step-by-step methodologies, example problems, and a poetic interlude for an active break.

Main Ideas and Concepts

Trigonometric Equations of Degree One: Equations involving trigonometric functions (sine, cosine, tangent, etc.) raised only to the first power (no higher exponents).
Periodic Nature of Trigonometric Functions: Because trig functions repeat values over intervals (periods), solutions are often infinite and expressed generally with an integer \( k \).
Use of Trigonometric Identities: Essential for simplifying and solving equations. Common identities used include:
- Pythagorean identities
- Reciprocal identities
- Quotient identities
- Double angle identities
Constant \( B \) and Period: The coefficient multiplying the variable inside trig functions affects the period of the function, which must be considered when finding all solutions.
General Solution Format: Solutions are expressed as \( x = \text{angle} + k \times \text{period} \), where \( k \in \mathbb{Z} \).

Methodology / Step-by-Step Approach to Solve First-Degree Trigonometric Equations

Identify the EquationDetermine which trigonometric functions are present.
Rewrite/Simplify the EquationUse identities or algebraic manipulation to express the equation in terms of a single trig function if possible.
Find Basic SolutionsSolve the simplified equation for the variable (angle) within one period.
Apply Identities if NeededUse trigonometric identities (Pythagorean, reciprocal, double angle, etc.) to simplify or transform the equation.
Consider the Range and PeriodicityUse the constant \( B \) to find the period of the function and express all solutions by adding multiples of the period.
Verify SolutionsSubstitute solutions back into the original equation to confirm correctness.

Detailed Problem-Solving Examples from the Video

1. Motivation Question

Prove: \( \tan^2 x = \frac{1 - 1}{\cos^2 x} \) (interpreted as \( \tan^2 x = \sec^2 x - 1 \))
Identities used:
- Pythagorean Identity
- Reciprocal Identity
Conclusion: The correct identities are Pythagorean and reciprocal.

2. Activity 1: Solve \( \sin 2x = 0 \)

Use periodicity of Sine Function.
Solutions:
- \( 2x = 0 + 2k\pi \)
- \( 2x = \pi + 2k\pi \)
Divide by 2 (the constant \( B \)) to get:
- \( x = k\pi \)
- \( x = \frac{\pi}{2} + k\pi \), \( k \in \mathbb{Z} \)

3. Activity 2: Solve \( \sin 2x - \sin x = 0 \)

Use Double Angle Identity: \( \sin 2x = 2 \sin x \cos x \)
Factor: \( \sin x (2 \cos x - 1) = 0 \)
Solve separately:
- \( \sin x = 0 \Rightarrow x = 0 + 2k\pi, \pi + 2k\pi \)
- \( 2 \cos x - 1 = 0 \Rightarrow \cos x = \frac{1}{2} \Rightarrow x = \frac{\pi}{3} + 2k\pi, \frac{5\pi}{3} + 2k\pi \)