3+siny = 3cos^2 y for 0<Y<360

I bet if you used

cos^2 Y= 1-Sin^2 Y

you would end up with a quadratic equation. If you are stuck, let x=SinY, and solve it that way.

thanks

To solve the equation 3 + sin(y) = 3cos^2(y) for 0 < y < 360 degrees, we will follow these steps:

Step 1: Rearrange the equation
We start by subtracting 3 from both sides of the equation:
sin(y) = 3cos^2(y) - 3

Step 2: Substitute sin(y) using the Pythagorean Identity
We know that sin^2(y) + cos^2(y) = 1, so we can rewrite the equation using this identity:
sin(y) = 3(1 - sin^2(y)) - 3

Step 3: Distribute and simplify
Distribute the 3 to both terms inside the parentheses:
sin(y) = 3 - 3sin^2(y) - 3

Step 4: Combine like terms
Combine the constants on the right-hand side:
sin(y) = -3sin^2(y)

Step 5: Divide by sin(y)
Divide both sides of the equation by sin(y):
1 = -3sin(y)

Step 6: Solve for sin(y)
Divide both sides of the equation by -3:
sin(y) = -1/3

Step 7: Find y
The value of sin(y) = -1/3 is true for two angles: approximately -19.47 degrees (-30 degrees) and approximately 199.47 degrees (210 degrees).

Therefore, the solution to the equation 3 + sin(y) = 3cos^2(y) for 0 < y < 360 degrees is y ≈ -19.47 degrees and y ≈ 199.47 degrees.

To solve the equation 3 + sin(y) = 3cos^2 (y) for 0 < y < 360, you can follow these steps:

Step 1: Rearrange the equation
Start by subtracting 3 from both sides of the equation:
sin(y) = 3cos^2(y) - 3

Step 2: Use the identity cos^2(y) = 1 - sin^2(y)
Replace cos^2(y) with its equivalent form:
sin(y) = 3(1 - sin^2(y)) - 3

Step 3: Distribute 3
Multiply 3 by each term inside the parentheses:
sin(y) = 3 - 3sin^2(y) - 3

Step 4: Combine like terms
Combine the constant terms on the right side of the equation:
sin(y) = -3sin^2(y)

Step 5: Divide by sin(y)
Divide both sides of the equation by sin(y):
1 = -3sin(y)

Step 6: Divide by -3
Divide both sides of the equation by -3:
1/-3 = sin(y)

Step 7: Solve for y
Take the inverse sine (arcsine) of both sides to solve for y:
y = arcsin(-1/3)

The solution to the equation is y = arcsin(-1/3).