find to the nearesr degree, all values of x between 0 degrees and 360 degrees that satisfy the equation 2sinx + 4 cos 2x =3
To find the values of x between 0 and 360 degrees that satisfy the equation 2sinx + 4cos2x = 3, we can use trigonometric identities and solve it algebraically. Here's how we can do it step by step:
1. Start by using the double angle formula for cosine: cos2x = 1 - 2sin^2(x).
Substitute the double angle formula into the original equation:
2sinx + 4(1 - 2sin^2(x)) = 3.
2. Distribute the 4 to the terms inside the parentheses:
2sinx + 4 - 8sin^2(x) = 3.
3. Rearrange the equation to form a quadratic equation:
-8sin^2(x) + 2sinx + 1 = 0.
4. Divide the equation by -1 to make the coefficient of the first term positive:
8sin^2(x) - 2sinx - 1 = 0.
5. This quadratic equation can be factored or solved using the quadratic formula. In this case, let's use the quadratic formula:
sinx = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 8, b = -2, and c = -1.
sinx = (2 ± √((-2)^2 - 4(8)(-1))) / (2(8)).
sinx = (2 ± √(4 + 32)) / 16.
sinx = (2 ± √36) / 16.
sinx = (2 ± 6) / 16.
6. Now, we have two possible values for sinx:
sinx = (2 + 6) / 16 = 8 / 16 = 1/2, or
sinx = (2 - 6) / 16 = -4 / 16 = -1/4.
7. To find the values of x, we need to find the angles whose sine values are 1/2 and -1/4. Use the inverse sine function (sin^(-1)) to find these angles, also known as arcsin:
x = sin^(-1)(1/2) and x = sin^(-1)(-1/4).
8. Use a calculator or trigonometric table to find these values:
x = 30°, 150°, and x = -14.48°, -165.52°.
9. However, we only want values between 0 and 360 degrees, so we discard the negative angles and keep the positive angles:
The values of x that satisfy the equation are x = 30°, 150°.
Therefore, the values of x between 0 and 360 degrees that satisfy the equation 2sinx + 4cos2x = 3 are 30° and 150°.