## CSETMathGuru: THE Site for Single Subject Math

**For Trigonometry, you must be able to**:

·Prove sin²x + cos²x = 1 (using Pythagoras' Theorem and a Unit Circle),

·Prove and Apply the Law of Sines and Cosines

·Apply

a)Half & Double-Angle Formulae: sin2x, cos2x, tan2x, sin ½ x, cos ½ x, tan ½ x

b)Addition-Subtraction Formulae: sin (x ± y), cos (x ± y), tan (x ± y)

c)(IF YOU HAVE TIME TO SPARE) Product-to-Sum and Sum-to-Product Formulae: sin x sin y, cos x cos y, sin x cos y, cos x sin y, sin x ± sin y, cos x ± cos y

·(IF YOU HAVE TIME TO SPARE) Prove of a)-c) above!

·Calculate trigonometric ratios of angles in degrees & radians (usually embedded into another Q)

·Solve fairly sophisticated trigonometric equations: involving preliminary factoring/ trigonometric or algebraic simplification/ use of quadratic formula, etc.

·Graph fairly sophisticated trigonometric functions of form y = A sin (Bx + C) + D, y = A cos (Bx + C) + D, y = A tan (Bx + C) + D or of their reciprocals (cosec, sec, cot: simply flip the base graphs!)

·Prove trigonometric identities: prove Left Hand Side = Right Hand Side

·Prove and Apply Demoivre's Theorem and Finding roots of complex #

·Use elementary trigonometry to find Areas and Perimeters of regular n-sided polygons (48-sided, 64-sided, etc.) or coordinates of vertices of polygons in the coordinate plane

**For Calculus, you must be able to**

·Calculate fairly sophisticated Algebraic and Trig. Limits: esp. those involving factorization/rationalizing the denominator/L'Hospital's Rule for 0/0, infinity/infinity and their variations)

·Prove the fundamental trigonometric limit using Sandwich Theorem: Limit x -> 0 sinx/x = 1

·Define Continuity of Functions and Conditions for Continuity: know Right Hand and Left Hand Limits

·Find the derivative of a function using First Principles (ie. using Limits and definition of a derivative)

·Distinguish between Continuity and Differentiability of functions: when is a function continuous but not differentiable, when does a function not have a derivative at a point; understand Right Hand and Left Hand Derivatives

·Apply Intermediate Value Theorems of functions and derivatives

·Apply fundamental Product/Quotient/Chain Rules for determining Derivatives of functions

·Perform Implicit & Parametric Differentiation

·Determine slopes and equations of tangent lines to graphs: understand the geometric interpretation of the derivative

·Solve Word Problems on Rates of Change

·Apply Mean Value Theorems: Rolle's, Lagrange's

·Graph rational/polynomial functions by a) x and y intercepts b) Vertical and Horizontal Asymptotes c) maxima /minima of function d) concavity/ convexity (first and second derivatives) e) End behavior

·Apply maxima-minima concepts to solve word problems about optimizing dimensions of figures/solids subject to constraints

·Define the Definite Integral (Sum of areas) and Calculate Definite Integral using Riemann Sums

·Apply Properties, Areas, Mean Value Theorem with respect to Definite Integrals

·Prove the Fundamental Theorem of Integral Calculus

·Find areas between curves

·Calculate SIMPLE algebraic integrals ONLY (using substitution)

·Determine Convergence of Infinite Series: using Comparison, Ratio, Root and other Tests

·Apply concepts relating to Taylor, Maclaurin Series