Solving and Analyzing the Trigonometric Equation cos θ sin θ √2 cos θ

In this article, we will detail the process of solving the trigonometric equation cos θ sin θ √2 cos θ. We will explore various methods to rearrange and analyze the given equation, using trigonometric identities to uncover the underlying relationships and derive the solutions.

Rearranging the Equation

To solve the equation cos θ sin θ √2 cos θ, we first rearrange the terms:

Cosine and sine terms are isolated on one side: The equation simplifies as follows:

(cos θ sin θ - √2 cos θ 0)

This simplifies to:

((sin θ - √2) cos θ 0)

Isolating sin θ

We isolate (sin θ) by dividing both sides by (cos θ), assuming that (cos θ ≠ 0):

(sin θ √2 cos θ)

Further simplifying, we get:

(tan θ √2 - 1)

Finding θ

To find the specific angle (θ), we take the arctangent of both sides:

(θ tan^{-1}(√2 - 1))

Verifying the Solutions

To verify our solution, we can substitute (θ tan^{-1}(√2 - 1)) back into the original equation:

Using Trigonometric Identities

An alternative method involves using trigonometric identities to express (cos θ sin θ) in a different form:

(cos θ sin θ √2 left( frac{1}{√2} cos θ frac{1}{√2} sin θ right))

This simplifies to:

((√2) left( cos θ sin θ right) left( frac{1}{√2} right))

Further simplifying:

(cos left(θ - 45°right) cos θ)

This implies:

(θ - 45° 2nπ) or (θ - 45° -2nπ)

for any integer (n).

The general solution can thus be expressed as:

(θ 45° 2nπ) or (θ 45° - 2nπ)

In conclusion, the equation (cos θ sin θ √2 cos θ) leads to a relationship involving the tangent function and can be expressed in terms of specific angles. The solutions can be found through trigonometric identities or rearrangement of the terms.