Difficulty: Foundational | Prerequisites: Calculus I and II trigonometry
This is a reference sheet for the trigonometric identities and derivative formulas you need to have at your fingertips for Calculus III. None of this is new material, but it appears constantly in curvature, arc length, and acceleration problems, and forgetting a derivative or identity mid-problem is the fastest way to lose marks.
Pythagorean identities
The three fundamental relationships between trig functions derived from sin²θ + cos²θ = 1. These are used constantly to simplify expressions inside radicals when computing arc length and curvature.
Inverse trigonometric functions
Functions that reverse the standard trig functions: arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹), etc. Their derivatives appear whenever you differentiate position or velocity functions that contain inverse trig expressions (as in the F = ma worked example on the cheat sheet).
These three Pythagorean identities from the cheat sheet are the ones you will use most:
sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cot²θ = csc²θ
The first identity is used constantly when simplifying |r'(t)| for curves involving sin and cos. For example, if r(t) = ⟨a cos t, a sin t, bt⟩, then |r'(t)|² = a² sin²t + a² cos²t + b² = a² + b², which simplifies to a constant thanks to sin²t + cos²t = 1.
The second and third identities appear when differentiating tan, sec, cot, and csc, and when simplifying expressions involving these functions.
All six, as listed on the cheat sheet:
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
A useful pattern to remember: the "co-" functions (cos, cot, csc) all have negative signs in their derivatives.
All six, as listed on the cheat sheet:
d/dx (arcsin x) = 1 / √(1 − x²)
d/dx (arccos x) = −1 / √(1 − x²)
d/dx (arctan x) = 1 / (1 + x²)
d/dx (arccot x) = −1 / (1 + x²)
d/dx (arcsec x) = 1 / (|x|√(x² − 1))
d/dx (arccsc x) = −1 / (|x|√(x² − 1))
The same "co-" pattern holds: arccos, arccot, and arccsc each have a negative sign relative to their counterpart.
These derivatives appear in Calculus III whenever a position or velocity component involves an inverse trig function. The cheat sheet's F = ma example uses arctan t in one component, requiring d/dx (arctan x) = 1/(1 + x²) to find the velocity.
Pythagorean identities:
sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cot²θ = csc²θ
Trig derivatives:
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
Inverse trig derivatives:
d/dx (arcsin x) = 1/√(1 − x²), d/dx (arccos x) = −1/√(1 − x²)
d/dx (arctan x) = 1/(1 + x²), d/dx (arccot x) = −1/(1 + x²)
d/dx (arcsec x) = 1/(|x|√(x² − 1)), d/dx (arccsc x) = −1/(|x|√(x² − 1))
Forgetting the negative sign on d/dx (cos x). It is −sin x, not sin x. The same error applies to cot and csc.
Confusing d/dx (arctan x) = 1/(1 + x²) with d/dx (arcsin x) = 1/√(1 − x²). Arctan has 1 + x² in the denominator (no square root); arcsin has 1 − x² under a square root.
Mixing up sec²x (the derivative of tan x) with tan²x. They differ by 1: sec²x = 1 + tan²x.
Applying the Pythagorean identity sin²θ + cos²θ = 1 but forgetting to use tan²θ + 1 = sec²θ when simplifying expressions involving tan and sec.
⚠️ These derivatives are not tested in isolation at the Calc III level, but they appear inside every curvature, arc length, and velocity computation. An error here propagates through the entire problem.
⚠️ The Pythagorean identity sin²θ + cos²θ = 1 is the single most-used simplification in arc length and curvature problems. If your |r'(t)| is not simplifying, check whether you can apply it.
⚠️ If a position component involves arctan or ln, expect its derivative to involve 1/(1 + t²) or 1/t, and plan your algebra accordingly.
Fill in the blank: d/dx (tan x) = ________. (sec²x)
True or false: d/dx (arccos x) = 1/√(1 − x²). (False. It is negative: −1/√(1 − x²).)
Fill in the blank: sin²θ + cos²θ = ________. (1)
True or false: The derivative of csc x is −csc x cot x. (True.)
Q: Simplify √(9 sin²t + 9 cos²t + 16).
A: 9 sin²t + 9 cos²t = 9(sin²t + cos²t) = 9. So √(9 + 16) = √25 = 5.
Q: Find d/dt (arctan(3t)).
A: By the chain rule: 1/(1 + (3t)²) × 3 = 3/(1 + 9t²).
Q: If r(t) = ⟨sin t, cos t, t⟩, what is |r'(t)|?
A: r'(t) = ⟨cos t, −sin t, 1⟩. |r'(t)| = √(cos²t + sin²t + 1) = √(1 + 1) = √2.
Q: What is d/dx (sec x tan x)? (Product rule practice.)
A: sec x tan x × tan x + sec x × sec²x = sec x tan²x + sec³x. This can also be written sec x(tan²x + sec²x).
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