5sin(2x)+4cos(x)=0
To solve the equation 5sin(2x) + 4cos(x) = 0, we can use trigonometric identities and algebraic manipulation.
Step 1: Rearrange the equation to have only one trigonometric function.
- Start by rewriting 5sin(2x) + 4cos(x) = 0 as 4cos(x) = -5sin(2x).
- Divide both sides of the equation by 4 to get cos(x) = (-5/4)sin(2x).
Step 2: Use a trigonometric identity to express sin(2x) in terms of cos(x).
- The double-angle identity for sine states that sin(2x) = 2sin(x)cos(x).
- Substitute this into the equation cos(x) = (-5/4)sin(2x) to obtain cos(x) = (-5/4)(2sin(x)cos(x)).
Step 3: Solve for cos(x) by canceling out common factors.
- Divide both sides of the equation by cos(x) to get 1 = (-5/2)sin(x).
- Divide both sides of the equation by (-5/2) to get -2/5 = sin(x).
Step 4: Find the values of x that satisfy the equation.
- Take the inverse sine (or arcsin) of both sides of the equation to isolate x. This gives x = arcsin(-2/5).
- The arcsin function returns the angle whose sine is the given value. Make sure to consider its domain, which is typically -π/2 to π/2.
- Use a calculator to approximate the value of arcsin(-2/5).
Therefore, the solution to the equation 5sin(2x) + 4cos(x) = 0 is x ≈ -0.4113 + kπ, where k is an integer.