find value of tan44degrees using integrals
MathMate answered this same question
http://www.jiskha.com/display.cgi?id=1289877322
To find the value of tangent (tan) 44 degrees using integrals, we can utilize the tangent's relationship to the derivative of the natural logarithmic function.
The derivative of the natural logarithmic function ln(x) is 1/x. Therefore, by integrating 1/x, we can find the integral of tangent which will help us evaluate tan(44 degrees).
Let's go through the steps:
1. Convert 44 degrees to radians: tan(x) is usually calculated using radians, so we need to convert 44 degrees to radians. To convert degrees to radians, multiply by π/180. Therefore, 44 degrees is equal to (44 * π) / 180 radians.
2. Write the integral: We want to find the integral of 1/cos(x). Since tan(x) = sin(x)/cos(x), we can rewrite the integral as the integral of sin(x) divided by cos(x).
3. Calculate the integral: To calculate the integral of sin(x)/cos(x), we can use a substitution technique. Let u = cos(x), which means du = -sin(x) dx. Rearranging the terms, we have -du = sin(x) dx.
Substituting these values into the integral, we get:
∫ (1/cos(x)) dx = ∫(u/u^2) (-du) = -∫(1/u) du
Now we can integrate -∫(1/u) du:
(-∫(1/u) du) = -ln|u| + C
4. Substitute back the original variable: Since we initially let u = cos(x), we have:
- ln|u| + C = -ln|cos(x)| + C
5. Evaluate at 44 degrees: Finally, substitute x = (44 * π) / 180 radians into the equation:
= -ln|cos((44 * π) / 180)| + C
Calculation: Plugging this expression into a calculator or software, the approximate value of -ln|cos((44 * π) / 180)| is -0.965+.
Note: The '+ C' denotes the constant of integration and can be ignored when evaluating a specific value.
Therefore, the value of tan(44 degrees) using integrals is approximately -0.965.