Section 6.1

• Know why radian measure is used instead of degrees.
• Be able to convert radian measures to degrees and degree measures to radians.
• Be very familiar with radian measures of common angles.
• Be able to calculate radian measures, arc lengths, sector areas, linear speeds, and angular speeds.
• Understand the difference between linear speed and angular speed.

• Know and understand the unit circle definitions of the six trigonometric functions.
• Be able to use the definitions to evaluate trigonometric functions at multiples of pi/2 or angles corresponding to given points.
• Be able to use the definitions to estimate trigonometric functions at any angle on a graph of a unit circle or to compare values of trigonometric functions.

• Be able to evaluate the trigonometric functions at standard angles in the first quadrant.
• Know the definition of reference angle.
• Be able to use reference angles to evaluate trigonometric functions at standard angles in any quadrant.

• Know the notational conventions for using trigonometric functions in algebraic expressions.
• Be able to correctly manipulate trigonometric functions in algebraic expressions.
• Know the basic identity relating sine and cosine and the basic identities defining tangent, cotangent, secant, and cosecant.
• Be able to use these basic identities to simplify algebraic expressions and to prove other identities.

• Know and understand the right triangle definitions of the six trigonometric functions.
• Be able to use the definitions to evaluate trigonometric functions from given right triangles.
• Know the basic identities relating sine and cosine of complementary angles.
• Be able to find trigonometric functions as expressions in a variable from given right triangles involving that variable.

• Know the definition of the trigonometric functions as functions of a real variable and the relations between them.
• Be able to derive the Pythagorean identities and use them to prove other identities.
• Know and be able to use the opposite angle identities and periodicity properties of sine and cosine to evaluate expressions.

• Know the definitions of period, periodic function, and amplitude.
• Know the properties of the graphs of sin(x) and cos(x).
• Be able to carefully graph sin(x) and cos(x) and to use their graphs to evaluate sin(x), cos(x), sin^(-1)(x), and cos^(-1)(x).

• Be able to graph Asin(Bx-C) and Acos(Bx-C).
• Know how the period, amplitude, and phase shift depend on A, B, and C and be able to use this to find period, amplitude, and phase shift for a given function.
• Given a graph of a function, be able to provide an equation for the graph of the form Asin(Bx) or Acos(Bx).
• Know how to use functions of the form Asin(Bx-C) and Acos(Bx-C) to model periodic data.

• Know the definition of simple harmonic motion and frequency.
• Be able to use functions of the form Asin(Bx-C) and Acos(Bx-C) to answer questions about applications involving simple harmonic motion.

• Know the domain, range, period, intercepts, and asymptotes of the graphs of tan(x), cot(x), csc(x), and sec(x).
• Be able to carefully graph tan(x), cot(x), csc(x), and sec(x) and functions related to these by shifts, stretches, and reflections.

• Know the addition formulas for sine, cosine, and tangent.
• Be able to derive "subtraction formulas" from the addition formulas.
• Be able to use the addition formulas to simplify expressions, compute quantities, and prove identities.

• Be able to derive the double-angle formulas for sine, cosine, and tangent from the addition formulas.
• Know the half-angle formulas for sine, cosine, and tangent.
• Know the alternative forms of the double-angle and half-angle formulas.
• Be able to use the double-angle and half-angle formulas to evaluate quantities and prove identities.

• Know and understand the derivation of the product to sum formulas.
• Know and understand the derivation of the sum to product formulas.
• Be able to use the product to sum and sum to product formulas to convert and simplify expressions and to prove identities.

• Understand the difference between a conditional equation and an identity.
• Be able to solve basic equations involving trigonometric functions, using inverse functions when appropriate.

• Know the definitions of the inverse sine, cosine, and tangent functions.
• Know the domain, range, and defining equations for the inverse sine, cosine, and tangent functions.
• Be able to use the inverse functions to evaluate quantities and simplify expressions.
• Be able to graph the inverse functions and closely related functions.

• Be able to use trigonometric functions to set up and solve problems in applications involving right triangles.
• Know and be able to derive the formula for the area of a triangle in terms of the lengths of two sides and the sine of the included angle.

• Know the Law of Sines and the Law of Cosines and why they are true.
• Be able to use the Law of Sines and the Law of Cosines to find lengths and angles in triangles.

• Know the definitions of scalar, vector, magnitude, and components.
• Understand which physical quantities should be represented by vectors and why.
• Understand the translation convention involved in equality of vectors.
• Know how to find the sum of two vectors geometrically and how to use this in application to forces and velocities.

• Know how to find the length of a vector and sum or difference of two vectors algebraically.
• Know how to find the components of a vector pointing from one point to another.
• Know and be able to use the algebraic properties of vector addition and scalar multiplication.

• Understand basic examples of parametric equations.
• Know the parametric equations of an ellipse (hence in particular, of a circle).
• Be able to find coordinates of points on a curve given by parametric equations.
• Be able to eliminate a parameter t from basic parametric equations and graph the resulting curve.

• Understand that polar coordinates can be used to specify a point in the plane.
• Be able to use the relations between polar and rectangular coordinates to convert quantities or equations in one form to the other.

• Be able to apply symmetry tests to equations in polar coordinates to help in graphing the equations.
• Be able to graph or identify basic polar equations.

• Understand how to give the trigonometric or polar form of a complex number.
• Be able to multiply and divide complex numbers using the polar forms.
• Know and be able to use DeMoivre's Theorem to find powers of complex numbers.