Solve the equation cos^2A = sin^2A + sinA when 0¨¬¡A¡360¨¬. Select all the solutions from the list below.
A. 330 degrees
B. 270 degrees
C. 150 degrees
D. 30 degrees
since cos^2 = 1-sin^2,
1-sin^2A - sin^2A - sinA = 0
2sin^2A + sinA - 1 = 0
(2sinA-1)(sinA+1) = 0
sinA = 1/2 or -1
A = 30,150,270
To solve the equation cos^2A = sin^2A + sinA, we can use the trigonometric identity cos^2A = 1 - sin^2A to rewrite the equation as 1 - sin^2A = sin^2A + sinA.
Let's simplify the equation and solve for sinA:
1 - sin^2A = sin^2A + sinA
1 = 2sin^2A + sinA
2sin^2A + sinA - 1 = 0
Now we have a quadratic equation in sinA. We can solve this equation by factoring or using the quadratic formula.
To factor the quadratic equation, we need two numbers that multiply to give -2 and add up to 1. The numbers that satisfy these conditions are 2 and -1:
(2sinA - 1)(sinA + 1) = 0
Setting each factor equal to zero, we get:
2sinA - 1 = 0 or sinA + 1 = 0
Solving the first equation, we have:
2sinA - 1 = 0
2sinA = 1
sinA = 1/2
This equation gives us the angle A = 30 degrees or A = 150 degrees.
Solving the second equation, we have:
sinA + 1 = 0
sinA = -1
This equation gives us the angle A = 270 degrees.
Therefore, the correct solutions are:
A. 30 degrees
B. 150 degrees
C. 270 degrees
So, the correct options from the given list are:
A. 330 degrees (not a solution)
B. 270 degrees (correct)
C. 150 degrees (correct)
D. 30 degrees (correct)