Question $2 x-1=2 \sin x+\cos x$

Studdy Solution

STEP 1

1. The equation $2x - 1 = 2 \sin x + \cos x$ involves both linear and trigonometric terms.
2. We will need to find a solution for $x$ that satisfies both sides of the equation.

STEP 2

1. Analyze the equation for possible simplifications or substitutions.
2. Use numerical or graphical methods to approximate solutions.
3. Verify the solution(s).

STEP 3

First, observe the structure of the equation. We have a linear term on the left and trigonometric terms on the right:
$2x - 1 = 2 \sin x + \cos x$
Consider if there are any trigonometric identities or substitutions that could simplify the equation. In this case, no direct simplification is apparent, so we proceed to numerical or graphical methods.

STEP 4

Since the equation involves trigonometric functions, it is often helpful to use numerical methods or graphing to find approximate solutions.
1. Graph the functions $y_1 = 2x - 1$ and $y_2 = 2 \sin x + \cos x$ .
2. Look for points of intersection, which correspond to the solutions of the equation.
Alternatively, use a numerical solver or iterative method (such as the Newton-Raphson method) to find approximate solutions for $x$ .

STEP 5

Verify the solution(s) found in Step 2 by substituting back into the original equation to ensure both sides are equal.
For example, if a solution $x = a$ is found, check:
$2a - 1 = 2 \sin a + \cos a$
If the equality holds, then $x = a$ is a valid solution.
Since this problem involves transcendental equations, exact symbolic solutions may not be easily obtainable, and numerical approximations are typically used.

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