Solve 8 sin(2 x) = 6 for the two smallest positive solutions A and B, with A < B
Give your answers accurate to at least two decimal places.
To solve the equation 8sin(2x) = 6, we need to isolate the variable x. Here's how we can do it step-by-step:
Step 1: Divide both sides of the equation by 8:
sin(2x) = 6/8
sin(2x) = 3/4
Step 2: Find the inverse sine (or arcsine) of both sides of the equation to eliminate the sine function:
2x = arcsin(3/4)
Step 3: Divide both sides of the equation by 2:
x = (1/2) * arcsin(3/4)
Now, we need to find the two smallest positive solutions A and B, with A < B. Let's evaluate x accurately to at least two decimal places:
Step 4: Calculate arcsin(3/4) using a calculator:
arcsin(3/4) ≈ 0.8481
Step 5: Substitute the value back into the equation to find the smallest positive solution:
A = (1/2) * 0.8481
≈ 0.4241
Step 6: To find the second positive solution, we need to add the period of the sine function, which is 2π. However, we also need to make sure x is still within the domain of the arcsine function (−π/2 ≤ x ≤ π/2). Since 2π is outside this range, we add 2π and then check if the result is still within the range.
x' = 0.4241 + 2π
≈ 6.5664
Since x' is outside the range of the arcsine function, we need to find the equivalent angle within this range. We do this by subtracting 2π until we obtain a value within the range.
x'' = 6.5664 - 2π
≈ -0.5922
Therefore, the second positive solution within the given range is:
B = (1/2) * -0.5922
≈ -0.2961
So, the two smallest positive solutions, accurate to at least two decimal places, are A ≈ 0.4241 and B ≈ -0.2961.
To solve the equation 8 sin(2x) = 6, we need to isolate the variable x.
Step 1: Divide both sides by 8:
sin(2x) = 6/8
sin(2x) = 3/4
Step 2: To solve for x, we need to take the inverse sine (also known as arcsine) of both sides. This will cancel out the sine function on the left side and leave us with x.
x = arcsin(3/4)
Step 3: Use a calculator to find the value of arcsin(3/4). The arcsin function gives you the angle whose sine is equal to the given value. In this case, we want the values of x.
Using a calculator, we find that:
x ≈ 0.8481 (rounded to four decimal places)
Step 4: Since the equation is periodic, we need to find the smallest positive solution A and then find the next positive solution B.
Examining the unit circle, we can see that the sine function repeats itself every 2π radians (or 360 degrees).
The smallest positive solution A occurs at x ≈ 0.8481.
To find the next positive solution B, we can add one period (2π) to the value of A:
B = A + 2π
B ≈ 0.8481 + 2π
B ≈ 0.8481 + 6.2832
B ≈ 7.1313 (rounded to four decimal places)
Therefore, the two smallest positive solutions A and B, with A < B, to the equation 8 sin(2x) = 6, accurate to at least two decimal places, are:
A ≈ 0.85
B ≈ 7.13
sin 2x = 3/4
2x = .848 or pi=.848
so, x = .424 or pi/2 - .424