How to Solve the Equation sin(2x) = cos(x)

TL;DR

To solve the equation sin(2x) = cos(x), use the double angle identity to rewrite sin(2x) as 2sin(x)cos(x). Rearrange the equation to 2sin(x)cos(x) - cos(x) = 0, then factor out cos(x) and solve for the values of x, resulting in x = π/2, 5π/6, and 3π/2.

Transcript

let's solve this trig equation we have sine of two  theta that's equal to cosine of theta what should   we do well we have two theta here right and this  is only theta so that's not good but this is 2   theta except the sign we do have the double angle  formula for sine right so let's use that this it's   the same as well 2 sine theta times cosine ... Read More

Key Insights

• 👨‍💼 The double angle formula for sine is used to simplify the trigonometric equation.
• 👻 Factoring out common terms allows for easier manipulation of the equation.
• ❓ Solving for cosine theta yields the values of 0 and PI/2.
• 👨‍💼 Solving for sine theta yields the value of 1/2.

Questions & Answers

Q: How is the double angle formula for sine used in solving the trigonometric equation?

The double angle formula is used to express sine of 2 theta as 2 sine theta times cosine theta. This allows us to simplify the equation and make all the angles the same to proceed with solving.

Q: Why is cosine theta subtracted on both sides of the equation?

By subtracting cosine theta on both sides, the right-hand side becomes 0, simplifying the equation and making it easier to factor out common terms.

Q: How are the values of cosine theta found in the equation?

The equation is factored to separate two terms: cosine theta and 2 sine theta minus 1. By setting each term equal to 0 individually, the values of cosine theta are found to be 0 and PI/2.

Q: How are the values of sine theta found in the equation?

By setting 2 sine theta minus 1 equal to 0, the value of sine theta is found to be 1/2. This is represented in a right triangle with a special 30-60-90 angle configuration.

Summary & Key Takeaways

• The video explains the steps to solve a trigonometric equation involving sine and cosine by using the double angle formula and factoring.

• The equation is simplified by subtracting cosine theta on both sides and factoring out cosine theta.

• The video then focuses on solving separately for the values of cosine theta and 2 sine theta minus 1.