Question

Sin(3π/2x) = -cos(x)
a) x = 1/2
b) x = 3/4
c) x = 1/4
d) x = 1/8

Answer 1

To solve the equation sin(3π/2x) = -cos(x), convert sin(3π/2x) using the identity sin(a) = -cos(a - π/2). Set the angles equal to each other and solve for x to get x = 1/4.

Step-by-step explanation:

To solve the equation sin(3π/2x) = -cos(x), we need to find the value of x that satisfies this equation. Here are the steps:

1. Start by converting sin(3π/2x) into an equivalent expression using the identity sin(a) = -cos(a - π/2). This gives us -cos(π/2 - 3π/2x) = -cos(x).
2. Since the left side and the right side are equal, we can set the angles equal to each other: π/2 - 3π/2x = x.
3. Solve the equation for x by simplifying: π/2 = 5π/2x. Divide both sides by 5π/2 to get x = 1/4.

The student asked the question: Which value of x satisfies the equation sin(3π/2x) = -cos(x)?

To solve this, we first note that sin(3π/2) is equal to -1 (as it corresponds to the y-coordinate on the unit circle at an angle of 3π/2 radians), and the equation simplifies to sin(x) = cos(x) when 3π/2x is at an angle where sine is equal to -1. This simplifies further to tan(x) = 1, since tan(x) is sin(x)/cos(x). The angles at which tan(x) = 1 within the range [0, 2π) are π/4 and 5π/4. Since the student provided options that only include positive angles less than 1, the only option that corresponds to π/4 is x = 1/4.

Therefore, the correct answer is c) x = 1/4.