MTH6131 Chap.6 Trigonometric and Inverse Trigonometric Derivatives
Trigonometric and Inverse Trigonometric Derivatives
One proof, six derivatives
The notes prove the derivative of sine from first principles. Expanding sin(x + Δx) with the addition formula and regrouping gives cos x times sin(Δx)/Δx plus sin x times (cos(Δx) − 1)/Δx. The two limits from the limits chapter are 1 and 0, so the derivative of sine is cosine.
The derivative of cosine is minus sine, and the other four follow with the quotient rule: tangent gives secant squared, cotangent gives minus cosecant squared, secant gives secant tangent, and cosecant gives minus cosecant cotangent.
Every function beginning with co carries a minus sign.
The chain factor carries the marks
Almost every item in the trigonometric exercises has an argument other than plain x, such as sin(4x − 5) or cos(3x/2), and the chain factor from that argument is the step most often dropped. The worked solutions write it out every time and then tidy the answer with a double-angle or product-to-sum identity.
Related-rates items with angles convert degrees to radians first, because every derivative formula assumes radians.
Inverse functions on restricted domains
Sine and cosine are not one to one, so they have inverses only after restriction: sine to the interval from −π/2 to π/2, cosine to the interval from 0 to π.
The derivative of arcsine comes from implicit differentiation of x = sin y, with the positive square root chosen because cosine is non-negative on the restricted interval. The derivative of arctangent needs no sign choice at all.
Two conventions for the inverse secant
The notes derive the derivative of arcsecant twice, because two range conventions are in use.
With one, the derivative is 1/(x√(x² − 1)); with the other, it is 1/(|x|√(x² − 1)). The two agree for x greater than 1 and differ in sign for x less than −1, so an answer should name the convention it uses.
What this chapter covers
- 01
Sine differentiated from first principles
- 02
The six trigonometric derivatives and their signs
- 03
The chain factor from the argument
- 04
Identities for tidying answers
- 05
Radians in related rates
- 06
Restricted domains and inverse functions
- 07
Arcsine, arccosine and arctangent derivatives
- 08
The two arcsecant conventions
Worked example · free
A secant of a square
- 2Differentiate the outer secant.
- 1Multiply by the derivative of the inner x².
- 1Find the zeros of the derivative.
Key terms
- Radian
- The unit of angle in which the derivative of sine is exactly cosine; every trigonometric derivative formula assumes it.
- Arcsine
- The inverse of sine restricted to the interval from minus pi over two to pi over two, defined for inputs between minus one and one.
- Arctangent
- The inverse of tangent restricted to the open interval from minus pi over two to pi over two, defined for every real input.
- Restricted Domain
- A part of a function's domain on which it is one to one, chosen so that an inverse function exists.
Trigonometric and Inverse Trigonometric Derivatives FAQ
Do I need to memorise all six trigonometric derivatives?
It is worth it, but only two need real memorising: sine differentiates to cosine and cosine to minus sine. The other four follow from the quotient rule in a line each. The pattern that every function whose name begins with co has a minus sign in its derivative catches most sign errors.
Why is the derivative of arcsine positive?
Implicit differentiation of x = sin y gives the derivative as 1 over cos y. Arcsine takes values between minus pi over two and pi over two, where cosine is never negative, so the square root is taken with the positive sign. The graph of arcsine is rising everywhere on its domain, which confirms it.
What happens if I use degrees instead of radians?
Every derivative formula acquires a factor of pi over 180, because the derivative of sine is cosine only when angles are measured in radians. In related-rates problems quoted in degrees per second, convert the rate to radians per second before differentiating, or the answer will be wrong by a factor of about 57.
Exam move
Make a two-column card with the six derivatives and the four inverse forms, then drill items whose arguments are not plain x until the chain factor appears without thinking. For every inverse function answer, write the restricted range next to it, because that range decides every sign.
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