If θ represents an angle such that sin2θ = tanθ - cos2θ, then sin θ - cosθ =
A. -√2
B. 0
C. 1
D. 2√2
What equation can I use to solve this problem?
sin2A + cos2A - sinA/cosA = 0
2sinAcosA + 1 - 2sin^2 A - sinA/cosA = 0
2sinAcos^2 A + cosA - 2sin^2 A cosA - sinA = 0
2sinAcosA(cosA - coA) + (cosA - sinA) =
(cosA - sinA)(2sinAcosA + 1) = 0
cosA - sinA = 0 or ......
got it! That's all they asked for.
cosA - sinA = 0
Oh...okay. Thanks for the help.
pls simplify cos2θ ÷ sinθ + cosθ
To solve this problem, we can use the trigonometric identity:
sin²θ + cos²θ = 1
By rearranging the given equation, we have:
sin²θ + cos²θ = tanθ - cos²θ
Combining like terms, we get:
2cos²θ = tanθ - sin²θ
Now, let's use another trigonometric identity:
tanθ = sinθ / cosθ
By substituting this into the equation, we have:
2cos²θ = (sinθ / cosθ) - sin²θ
To simplify further, let's multiply both sides by cosθ:
2cos³θ = sinθ - cosθ * sin²θ
Using the trigonometric identity sin²θ = 1 - cos²θ, we can rewrite the equation as:
2cos³θ = sinθ - cosθ * (1 - cos²θ)
Expanding the equation, we have:
2cos³θ = sinθ - cosθ + cos³θ
Combining like terms, we get:
2cos³θ - cos³θ = sinθ - cosθ
Simplifying further, we have:
cos³θ = sinθ - cosθ
Now, we can use another trigonometric identity:
sinθ = √(1 - cos²θ)
By substituting this into the equation, we have:
cos³θ = √(1 - cos²θ) - cosθ
To solve this equation, we need to use a numerical method such as Newton's method or the bisection method.
However, the question asks for the value of sinθ - cosθ, not the exact value of θ. So instead of solving for θ, we can find the value of sinθ - cosθ using answer choices A, B, C, or D.
Let's substitute the answer choices into sinθ - cosθ and see which one satisfies the equation.
A. -√2:
sinθ - cosθ = -√2 - (-√2) = 0
B. 0:
sinθ - cosθ = 0 - 0 = 0
C. 1:
sinθ - cosθ = 1 - 1 = 0
D. 2√2:
sinθ - cosθ = 2√2 - (√2) = √2
Therefore, the correct answer is B. 0.
Note: In some cases, it may be necessary to solve the equation to find the exact value of θ. In this particular problem, however, we can determine the answer without finding the exact value of θ.