Question

3sin^2 x - cos x +3=0

Answer 1

3(1-cos^2x)+cos^2x-2=0
3-3cos^2x+cos^2x-2=0
-2cos^2x=-1
cos^2x=1/2
1)cosx=sqrt(2)/2
x=+-pi/4+2pi*n
2)cosx=-sqrt(2)/2
x=+-3pi/4+2pi*n
Ответ: x=+-pi/4+2pi*n; x=+-3pi/4+2pi*n

Answer 2

To solve the equation 3sin^2(x) - cos(x) + 3 = 0, we can use trigonometric identities to simplify it and then find the values of x that satisfy the equation. Here's the step-by-step process:

Step 1: Identify the trigonometric identities we can use:

• Pythagorean Identity: sin^2(x) + cos^2(x) = 1

Step 2: Substitute sin^2(x) from the Pythagorean Identity into the equation: 3(1 - cos^2(x)) - cos(x) + 3 = 0

Step 3: Expand the equation: 3 - 3cos^2(x) - cos(x) + 3 = 0

Step 4: Combine like terms: -3cos^2(x) - cos(x) + 6 = 0

Step 5: To make it easier, let's introduce a substitution: Let y = cos(x). The equation becomes: -3y^2 - y + 6 = 0

Step 6: Solve the quadratic equation: Now, we need to solve the quadratic equation for y. We can do this by factoring or using the quadratic formula. Let's use the quadratic formula: The quadratic formula is given by: y = (-b ± √(b^2 - 4ac)) / 2a

where a = -3, b = -1, and c = 6.

y = (1 ± √(1 - 4(-3)(6))) / 2(-3) y = (1 ± √(1 + 72)) / -6 y = (1 ± √73) / -6

Step 7: Find the values of x: Now that we have the values of y, we can find the corresponding values of x using the inverse cosine function (arccos):

For y = (1 + √73) / -6: x = arccos((1 + √73) / -6)

For y = (1 - √73) / -6: x = arccos((1 - √73) / -6)

Please note that arccos returns values in radians. If you need the solutions in degrees, you can convert the radians to degrees using the formula: degrees = radians * (180 / π).