Trigonometric Calculus when Angles are in Degrees
Today’s blog is a quickie.
The preferred angle measure in calculus is the radian. However, a lot of applications, including geometry, astronomy, engineering, and construction, use degrees.
An approach is to convert everything to radians before proceeding. Another approach is to remember that x radians = x° * π / 180, and use the conversion factor.
Derivatives
d/dx sin( x° )
Now all calculus calculations must have radians.
d/dx sin( x * π / 180)
= π / 180 * cos (x * π / 180)
= π / 180 * cos(x°)
Similarly – remember the angle considered is in DEGREES:
d/dx sin(x°) = π / 180 * cos(x°) |
d/dx csc(x°) = - π / 180 * csc(x°) * cot(x°) |
d/dx cos(x°) = - π / 180 * sin(x°) |
d/dx sec(x°) = π / 180 * tan(x°) * sec(x°) |
d/dx tan(x°) = π / 180 * sec(x°)^2 |
d/dx cot(x°) = -π / 180 * csc(x°) |
Integration
Now let’s try integration.
∫( sin(x°) dx)
= ∫( sin(x * π / 180)) dx
= 180 / π * ∫(π / 180 * sin(x * π / 180)) dx
= 180 / π * -cos(x * π / 180) + C
= -180 / π * cos(x°) + C
Similarly:
∫ sin(x°) dx = -180 / π * cos(x°) + C |
∫ cos(x°) dx = 180 / π * sin(x°) + C |
∫ tan(x°) dx = -180 / π * ln(cos(x°)) + C |
Use caution when using calculators. A lot of calculators when using calculus in degree mode get it correct but its’ always good to verify.
Eddie
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