Trigonometry and Calculus
Trigonometry and calculus
Know the unit circle, radian measure, graphs and transformations of sine, cosine, and tangent, restricted domains of inverse trigonometric functions, and identities, including Pythagorean, sum and difference, and double-angle identities. Solve triangles with the law of sines, law of cosines, and area = ½ab sin C. In calculus, understand limits, continuity, the derivative definition and rules (power, product, quotient, chain), applications such as slope, extrema, optimization, and related rates, plus antiderivatives, definite integrals, the Fundamental Theorem of Calculus, and area.
Traps
- Calculus derivative and integral formulas require radians; degree mode breaks them.
- A limit can exist where the function is undefined; continuity requires the limit to equal the function value.
- Differentiability implies continuity, but continuity does not imply differentiability at corners or cusps.
- Inverse sine returns values only in [−π/2, π/2].
- Indefinite integrals need +C; definite integrals may be negative because they measure signed area.
Worked example
Differentiate f(x) = x² sin x. By the product rule, f′(x) = 2x sin x + x² cos x. Rule of thumb: identify the outer structure first-product, quotient, or composition-then apply the matching rule from the outside in.