Trigonometric Functions --------------------------- Goals for Section 4.1 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Determine an angle's measurement in radians given either the arc length or radius of an arc. * Place an angle in standard position given a verbal description. * Determine the terminal or the initial side of a sector. * Recognize whether an angle is positive, negative or zero. * Recognize when two angles have coterminal sides. * Determine if an angle is acute, obtuse, or a right angle. * Determine if an angle is complimentary or a supplementary angle. * Given any two values of a sectors radius, angle, or arc length determine the value of the remaining quantity. * Determine the area of a sector given the radius and angle. * Given any two values of a sectors radius, angle, or arc length determine the value of the area of the sector. * Recognize when a given angle represents more than one complete rotation around a circle. Goals for Section 4.2 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Determine the coordinate of a point on the unit circle given either the *x* or the *y* value for the coordinate. * Given the radian measurement of an angle determine the approximate location of the corresponding point on the unit circle and determine which quadrant the point is in. * Determine the point on the unit circle when the corresponding angle is a multiple of :math:`\frac{\pi}{4}`. * Use the definition of trigonometric functions to determine the values of the functions given a coordinate on the unit circle. * State the domain and range of the trigonometric functions. * State the fundamental trigonometric identities. * Determine if a basic function is periodic and explain their conclusion in terms of the definition of a periodic function. * Determine the period of a trigonometric function. * Use a calculator to approximate trigonometric functions. Goals for Section 4.3 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Determine values of trigonometric functions given right triangles. * Use the Pythagorean identity to determine the value of one trigonometric function in terms of another. * Determine the values of trigonometric functions using multiple right triangles. * Translate a written description of a problem into a trigonometric formulation Goals for Section 4.4 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Determine subsets of the domain where sine/cosine are increasing and where they are decreasing. * Determine whether a trigonometric function will increase or decrease given a specific angle. * Determine the reference angle of a given angle in any of the four quadrants. Goals for Section 4.5 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Determine the range and domain of the sine and cosine functions. * Determine the amplitude, period, and phase of a trigonometric function given a written or graphical representation. * Determine the vertical shift and scale of a trigonometric function given a written or graphical representation. * Determine the horizontal shift and scale of a trigonometric function given a written or graphical representation. * Graph a sine or cosine wave given a formula or written description. * Determine the formula for a sine or cosine wave given a written description * Determine the formula for a sine or cosine wave given the graph or a written description of the function. * Determine the period of a sine or cosine wave given the graph or a written description of the function. Goals for Section 4.7 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Determine the restriction of the domain of a function in order to define an inverse. * State the canonical domain restrictions for sine, cosine, and tangent functions. * Use the definition of the arcsine, arccosine, and the arctangent functions to solve for one value in a trigonometric expression. * Sketch the graphs of the arcsine, arccosine, and the arctangent functions. * Compose trigonometric and inverse trigonometric functions to transform a trigonometric expression into an equivalent form that allows for direct algebraic manipulation to isolate a value in the domain of a trigonometric function in the original expression. * Determine equivalent expressions for the compositions of trigonometric and inverse trigonometric functions so that the results do not include any trigonometric functions. Goals for Section 5.1 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * State fundamental identities. * State the Pythagorean identitites. * Verify that a given expression is true. * Using quotient identities * Bringing together expressions using a common denominator. * Factoring expressions. * Using basic properties of trigonometric functions. (E.g.: :math:`\sin(-\theta)=-\sin(\theta)`.) * Solve for values within expressions that involve logarithms. * Transform algebraic expressions by substituting trigonometric functions. Goals for Section 5.2 of the book ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Students should be able to do each of the following: * Use the sum and difference formulas for the sine function to solve for values within a trigonometric expression. * Use the sum and difference formulas for the cosine function to solve for values within a trigonometric expression. * Verify identities that make use of sum and difference formulas. * Combine the sum and difference formulas with inverse trigonometric functions.