Calculus I - Week 4
Derivatives of Trigonometric Functions
This week covers the derivatives of all six trigonometric functions, their derivations using limits, and applications in physics involving oscillation, waves, and circular motion. Students practice chaining trig derivatives with the chain rule.
Learning Objectives
- Derive the derivative of sin(x) using the limit definition. - State and apply the derivatives of all six trig functions. - Combine trig derivatives with the chain, product, and quotient rules. - Differentiate functions like sin(2x), cos(x^2), and tan(3x). - Apply trig derivatives to find slopes and tangent lines. - Recognize trig derivatives in physics contexts (waves, oscillations).
1. Why Trig Functions Matter
Trigonometric functions describe waves, oscillations, and circular motion. The position of a pendulum, the height of a wave, and the voltage in an AC circuit all involve sin and cos. To work with these in calculus, we need their derivatives.
2. Derivative of sin(x)
Using the limit definition and two special limits: lim h->0 sin(h)/h = 1 lim h->0 (cos(h) - 1)/h = 0 We can show: d/dx [sin(x)] = cos(x) This means the slope of y = sin(x) at any point x equals cos(x).
3. Derivative of cos(x)
Similarly: d/dx [cos(x)] = -sin(x) Notice the negative sign. Where cos is decreasing (between 0 and pi), its slope is negative, which matches -sin(x) > 0 on (0, pi)... wait — let's be careful. sin(x) is positive on (0, pi), so -sin(x) is negative on (0, pi). And cos is indeed decreasing from 1 to -1 on (0, pi). The negative sign is correct.
4. All Six Trig Derivatives
d/dx [sin(x)] = cos(x) d/dx [cos(x)] = -sin(x) d/dx [tan(x)] = sec^2(x) d/dx [csc(x)] = -csc(x)cot(x) d/dx [sec(x)] = sec(x)tan(x) d/dx [cot(x)] = -csc^2(x) The top three (sin, cos, tan) are the most important. The bottom three (csc, sec, cot) come from the quotient rule applied to their definitions.
5. Chain Rule with Trig Functions
When the argument is not just x, apply the chain rule. d/dx [sin(g(x))] = cos(g(x)) * g'(x) Examples: - d/dx [sin(2x)] = cos(2x) * 2 = 2cos(2x) - d/dx [cos(x^2)] = -sin(x^2) * 2x = -2x sin(x^2) - d/dx [tan(3x)] = sec^2(3x) * 3 = 3sec^2(3x)
6. Product Rule with Trig Functions
Example: Differentiate h(x) = x * sin(x). f(x) = x, f'(x) = 1. g(x) = sin(x), g'(x) = cos(x). h'(x) = 1 * sin(x) + x * cos(x) = sin(x) + x cos(x). Example: Differentiate h(x) = x^2 * cos(x). h'(x) = 2x cos(x) + x^2(-sin(x)) = 2x cos(x) - x^2 sin(x).
7. Quotient Rule with Trig Functions
Example: Differentiate h(x) = sin(x) / x. f = sin(x), f' = cos(x). g = x, g' = 1. h'(x) = [cos(x) * x - sin(x) * 1] / x^2 = (x cos(x) - sin(x)) / x^2. Example: Differentiate tan(x) = sin(x)/cos(x) using the quotient rule. h'(x) = [cos(x)*cos(x) - sin(x)*(-sin(x))] / cos^2(x) = [cos^2(x) + sin^2(x)] / cos^2(x) = 1 / cos^2(x) = sec^2(x).
8. Tangent Lines to Trig Curves
Example: Find the tangent line to y = sin(x) at x = pi/2. Point: (pi/2, sin(pi/2)) = (pi/2, 1). Slope: y' = cos(pi/2) = 0. The tangent line is horizontal: y = 1. Example: Find the slope of y = cos(x) at x = pi/3. y' = -sin(x), so slope = -sin(pi/3) = -sqrt(3)/2.
9. Physics: Simple Harmonic Motion
A mass on a spring has position: x(t) = A cos(wt) where A is the amplitude and w is the angular frequency. Velocity: v(t) = x'(t) = -Aw sin(wt) Acceleration: a(t) = v'(t) = -Aw^2 cos(wt) = -w^2 x(t) This last equation, a = -w^2 x, is the defining equation of simple harmonic motion.
10. Common Mistakes
- Writing d/dx[cos(x)] = sin(x) (missing the negative sign). - Forgetting to multiply by the inner derivative in chain rule applications. - Confusing sec^2(x) and sec(x)tan(x) — remember tan's derivative is sec^2(x). - Writing d/dx[sin^2(x)] as 2sin(x) without the chain rule multiply — correct answer is 2sin(x)cos(x).