Derivatives of Trigonometric Functions: Notes and FAQs

The six trigonometric functions each have a clean, memorable derivative. Together they form the toolkit for differentiating anything involving angles, waves, and oscillations: d/dx sin x = cos x, d/dx cos x = −sin x, and four more for tangent, cotangent, secant, and cosecant.

These matter enormously because trigonometry describes periodic behavior everywhere: sound waves, alternating current, pendulums, and planetary orbits. Velocity is the derivative of position, so differentiating a sinusoidal position immediately demands these formulas. They are also the gateway to differentiating inverse trigonometric functions.

Key Concepts

• The big two: the derivative of sin x is cos x; the derivative of cos x is −sin x (note the minus sign).
• The other four: tan x → sec²x, cot x → −csc²x, sec x → sec x tan x, csc x → −csc x cot x.
• Radians required: all these formulas assume x is measured in radians. In degrees they gain messy extra factors.
• Chain rule combinations: most exam problems nest trig functions, like sin(2x), so the chain rule's extra factor appears.

Formulas

• d/dx sin x = cos x.
• d/dx cos x = −sin x.
• d/dx tan x = sec²x.
• d/dx cot x = −csc²x.
• d/dx sec x = sec x · tan x.
• d/dx csc x = −csc x · cot x.

Worked Example

Differentiate y = sin(2x) + cos x and evaluate y'(π/6).

Handle each term. For sin(2x), the chain rule applies: the outer derivative is cos(2x) and the inner derivative of 2x is 2, giving 2cos(2x). For cos x, the derivative is −sin x. So y' = 2cos(2x) − sin x. Now evaluate at x = π/6: the angle 2x becomes 2·π/6 = π/3. We know cos(π/3) = 1/2 and sin(π/6) = 1/2. Substituting: y'(π/6) = 2·(1/2) − 1/2 = 1 − 1/2 = 1/2. The answer is 0.5, obtained exactly using standard angle values.

Common Misconceptions

• The derivative of cos x is sin x. It is −sin x. The minus sign is the single most dropped sign in trigonometry.
• Degrees work just as well. They do not. These derivative formulas are proved using lim(x → 0) sin x/x = 1, which holds only in radians.
• sec²x and 1/cos²x are different answers. They are identical — sec x means 1/cos x, so both forms are correct.

Real-World Applications

• Simple harmonic motion: a pendulum's velocity comes from differentiating its sinusoidal displacement.
• Alternating current: voltage and current in AC circuits are sines and cosines; their rates of change use these derivatives.
• Wave physics: differentiating wave functions gives particle velocities in sound and light.
• Engineering vibrations: analyzing bridges and machinery under oscillatory loads.

In summary, memorize the six trig derivatives — especially the minus signs on cos, cot, and csc — always work in radians, and pair them with the chain rule for nested angles. They turn wave and oscillation problems into straightforward algebra.