sinx/1 cosx sinx/1-cosx=4 ,x lies in 1st quadrant then value of x is
To find the value of x in the equation sinx/(1-cosx) = 4, where x lies in the first quadrant, we can follow these steps:
Step 1: Simplify the equation
sinx/(1-cosx) = 4
Multiply both sides of the equation by (1-cosx):
sinx = 4(1-cosx)
Expand the right side of the equation:
sinx = 4 - 4cosx
Step 2: Determine the values of sin(x) and cos(x) in the first quadrant.
In the first quadrant, both sin(x) and cos(x) are positive. So, we can rewrite the equation as:
sinx = 4 - 4cosx
Since sin(x) is positive, we can rearrange the equation to isolate cos(x):
4cosx = 4 - sinx
cosx = (4 - sinx)/4
Step 3: Use the Pythagorean identity to rewrite cos(x) in terms of sin(x).
The Pythagorean identity states that:
sin^2(x) + cos^2(x) = 1
Rearranging this equation, we get:
cos^2(x) = 1 - sin^2(x)
Since cos(x) is positive in the first quadrant, we can take the positive square root:
cos(x) = sqrt(1 - sin^2(x))
Substituting this expression for cos(x) into the equation from step 2, we have:
sqrt(1 - sin^2(x)) = (4 - sinx)/4
Step 4: Solve the equation for sin(x).
Squaring both sides of the equation to eliminate the square root:
1 - sin^2(x) = (4 - sinx)^2/16
Multiplying both sides of the equation by 16 to eliminate the denominator:
16 - 16sin^2(x) = (4 - sinx)^2
Expanding the square on the right side:
16 - 16sin^2(x) = 16 - 8sinx + sin^2(x)
Rearranging and simplifying the equation:
17sin^2(x) - 8sinx = 0
Factoring out the common factor of sin(x):
sinx(17sinx - 8) = 0
Now we have two equations to solve:
1) sinx = 0
2) 17sinx - 8 = 0
Solving equation 1:
sinx = 0
x = 0 (since sin(0) = 0)
Solving equation 2:
17sinx - 8 = 0
sinx = 8/17
Taking the inverse sine of both sides:
x = arcsin(8/17)
Calculating the value of x using a calculator or approximate values, we find:
x ≈ 0.987 (rounded to three decimal places)
Therefore, the value of x in the first quadrant that satisfies the equation sinx/(1-cosx) = 4 is approximately 0.987.