Which one of the following angles is not a solution of 2cos^2 θ=1+sin θ
A: 90º B: 180º
C:270º
D:450º
E: none of the above
You could just sub in each of the given choices, or
you could just solve it:
2cos^2 θ = 1+sinθ
2(1 - sin^2 θ) - sinθ - 1 = 0
-2sin^2 θ - sinθ + 1 = 0
2sin^2 θ + sinθ - 1 = 0
(2sinθ - 1)(sinθ + 1) = 0
sinθ = 1/2 or sinθ = -1
θ = 30°, 150°, 270°
so ....
To find out which angle is not a solution of the given equation, we need to substitute each angle into the equation and check if it satisfies the equation.
Let's substitute each angle into the equation:
For angle A: θ = 90º
2cos^2(90º) = 1 + sin(90º)
2(0)^2 = 1 + 1
0 ≠ 2
For angle B: θ = 180º
2cos^2(180º) = 1 + sin(180º)
2(-1)^2 = 1 + 0
2 = 1
For angle C: θ = 270º
2cos^2(270º) = 1 + sin(270º)
2(0)^2 = 1 + (-1)
0 ≠ 0
For angle D: θ = 450º
2cos^2(450º) = 1 + sin(450º)
2(0)^2 = 1 + 1
0 ≠ 2
From the calculations, we can see that angle C (270º) is the only angle that is not a solution of the equation.
Therefore, the answer is C: 270º.
To determine which one of the listed angles is not a solution of the equation 2cos^2 θ = 1 + sin θ, we can use substitution and simplification.
First, let's substitute each given angle into the equation and see if it holds true.
A: 90º
Substituting θ = 90º into the equation:
2cos^2 90º = 1 + sin 90º
Since cos 90º = 0 and sin 90º = 1, the equation becomes:
2(0)^2 = 1 + 1
This simplifies to:
0 = 2
Since the equation is not satisfied, angle 90º (option A) is not a solution.
B: 180º
Substituting θ = 180º into the equation:
2cos^2 180º = 1 + sin 180º
Since cos 180º = -1 and sin 180º = 0, the equation becomes:
2(-1)^2 = 1 + 0
This simplifies to:
2 = 1
Since the equation is not satisfied, angle 180º (option B) is not a solution.
C: 270º
Substituting θ = 270º into the equation:
2cos^2 270º = 1 + sin 270º
Since cos 270º = 0 and sin 270º = -1, the equation becomes:
2(0)^2 = 1 + (-1)
This simplifies to:
0 = 0
Since the equation is satisfied, angle 270º (option C) is a solution.
D: 450º
Substituting θ = 450º into the equation:
2cos^2 450º = 1 + sin 450º
Since cos 450º = 0 and sin 450º = -1, the equation becomes:
2(0)^2 = 1 + (-1)
This simplifies to:
0 = 0
Since the equation is satisfied, angle 450º (option D) is a solution.
Based on our analysis, angles 90º (option A) and 180º (option B) do not satisfy the given equation. Therefore, the answer is E: none of the above.