How to Solve the Trigonometric Equation cos(2θ)=cos(θ)

TL;DR

To solve the equation cos(2θ) = cos(θ), use the double angle formula to transform it into a quadratic equation: 2cos²(θ) - cos(θ) - 1 = 0. Factor the equation, set the factors to zero, and find the solutions, which involve analyzing the unit circle to determine angles for cosine values of -1/2 and 1.

Transcript

okay let's solve this trig equation we have cosine  two theta is equal to cosine theta so what's the   trouble you should tell me that the angles are  different right this is two data this is only   data so what should we do look at this one we have  a double angle formula for cosine right which ones   you'll use well we have the cosine theta here ... Read More

Key Insights

• 💁 Trig equations can be solved using the double angle formula for cosine, which transforms the equation into a quadratic form.
• 🧑‍🏭 Factoring the quadratic equation is an essential step in solving trig equations.
• 🗯️ Understanding the unit circle and special right triangles helps determine the solutions for trig equations involving cosine.

Questions & Answers

Q: What is the initial equation in the video?

The initial equation is cosine(2θ) = cosine(θ), where θ represents an angle.

Q: Why do we use the double angle formula for cosine in this case?

We use the double angle formula because the equation involves both cosine squared and cosine to the first power, and we need to simplify it into a quadratic equation.

Q: How do we factor the quadratic equation in terms of cosine theta?

We use the take-title factoring method, which involves finding two values that multiply to give the coefficient of cosine squared and sum to give the coefficient of cosine.

Q: What are the solutions to the equation?

The solutions are θ = 2π/3 and θ = 4π/3, considering the restrictions mentioned in the video.

Summary & Key Takeaways

• The content shows how to solve a trig equation involving two different angles using the double angle formula for cosine.

• The equation is transformed into a quadratic equation, and the steps to factor it are explained.

• The factors are set equal to zero, and the solutions for cosine theta are determined by considering the unit circle and special right triangles.