Solving the Trigonometric Equation: 6cosx - 8sinx 5

${title}

When dealing with the equation 6cosx - 8sinx 5, the primary goal is to solve for the variable x rather than proving its equivalence to a constant. This is a common type of trigonometric equation that can be approached systematically using basic algebraic and trigonometric identities.

Step 1: Standardize the Equation

The first step is to standardize the equation, allowing us to work with a single trigonometric function. Notice that the coefficients of cosx and sinx can be expressed as the numerator of a fraction with a common denominator. Let's set this up:

6cosx - 8sinx 5

This can be rewritten as:

10left(frac{6}{10}cosx - frac{8}{10}sinxright) 5

Simplifying further, we get:

10left(frac{3}{5}cosx - frac{4}{5}sinxright) 5

Dividing both sides by 10, we have:

left(frac{3}{5}cosx - frac{4}{5}sinxright) frac{1}{2}

Step 2: Using Trigonometric Identities

To proceed, we can use the concept of a linear combination of sine and cosine. Let's rewrite the left-hand side as a single trigonometric function. Recall that any such expression can be written as R sin(x θ) or R cos(x - θ), where R is the amplitude and θ is the phase shift.

The amplitude R is given by:

R sqrt{left(frac{3}{5}right)^2 left(frac{4}{5}right)^2} sqrt{frac{9}{25} frac{16}{25}} sqrt{frac{25}{25}} 1

This simplifies the equation to:

1 cdot cos(x - theta) frac{1}{2}

Thus, the equation becomes:

cos(x - theta) frac{1}{2}

Step 3: Solving for x

From trigonometric tables or identities, we know that:

costheta frac{1}{2}

This happens when:

theta 2npi pm frac{pi}{3}

Therefore, we can write:

x - theta 2npi pm frac{pi}{3}

So the solution for x is:

x 2npi pm frac{pi}{3} theta

Given that:

theta tan^{-1}left(frac{4}{3}right)

The final solutions for x are:

x 2npi pm frac{pi}{3} - tan^{-1}left(frac{4}{3}right)

where n in mathbb{Z}.

Conclusion

It is important to understand that 6cosx - 8sinx 5 is not an identity but an equation, and thus cannot be proven as such. Instead, we solve it for x, as demonstrated above. This process involves standardizing the equation, using trigonometric identities, and solving for the variable x.

Keywords: Trigonometric Equations, Solving Trigonometric Equations, Proving Trigonometric Identities