Question

5sin^2x+3sinx*cosx=4*1​

Answer 1

Ответ:

Ответ во вложении.

Answer 2

To solve the trigonometric equation 5sin^2(x) + 3sin(x)*cos(x) = 4, we'll need to use trigonometric identities and algebraic manipulations. Let's go step-by-step:

Step 1: Use the identity sin(2x) = 2sin(x)cos(x) to rewrite the term 3sin(x)*cos(x): 5sin^2(x) + 3sin(x)*cos(x) = 4 5sin^2(x) + 2sin(2x) = 4

Step 2: Move all terms to one side of the equation to obtain a quadratic equation: 5sin^2(x) + 2sin(2x) - 4 = 0

Step 3: Make a substitution to simplify the equation: Let's make a substitution, y = sin(x), then sin(2x) = 2sin(x)cos(x) = 2y*cos(x). We can express sin^2(x) in terms of y as sin^2(x) = y^2.

The equation becomes: 5y^2 + 2y - 4 = 0

Step 4: Solve the quadratic equation for y: To solve the quadratic equation, we can use the quadratic formula: y = (-b ± √(b^2 - 4ac)) / 2a

where a = 5, b = 2, and c = -4.

y = (-(2) ± √(2^2 - 4 * 5 * -4)) / 2 * 5 y = (-2 ± √(4 + 80)) / 10 y = (-2 ± √84) / 10

Now, we have two possible values for y:

1. y = (-2 + √84) / 10 ≈ 0.4116
2. y = (-2 - √84) / 10 ≈ -0.8116

Step 5: Find x from the values of y: Since y = sin(x), we need to find the corresponding values of x. To do that, we use the inverse sine function (arcsin) on the calculated values of y:

1. If y = sin(x), then x = arcsin(0.4116) ≈ 24.18 degrees (or 0.422 radians)
2. If y = sin(x), then x = arcsin(-0.8116) ≈ -53.45 degrees (or -0.933 radians)

So, the solutions for the equation are approximately: x ≈ 24.18 degrees or 0.422 radians, and x ≈ -53.45 degrees or -0.933 radians.

Note: There could be more solutions due to the periodic nature of trigonometric functions. The above solutions are for one complete cycle of the sine function.