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Course Correlation to Virginia Standards of Learning Name of Provider: York County School Division Name of Course: Trigonometry URL for Course Syllabus: http://yorkcountyschools.org/virtualLearning/default.aspx Last Revision Date: 2010-11 (Courses revised quarterly as needed) Trigonometry SOL Standard: How does the course content address this standard? (Please refer to the syllabus posted on your Web site.) T.1 The student, given a point other than the Trigonometric Functions origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles. T.2 T.3 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions. The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa. Basic Concepts and Terminology o Recognize and draw an angle in standard position. – Calculate the values angles coterminal to a given angle in standard position. Radians o Recognize various systems used to measure angles, including degrees, radians, gradians, and mils. – Use a conversion factor to convert from radians to degrees and vice versa without using a calculator. Right Triangle Trigonometry (acute angles) o Define the six trigonometric functions of an angle in a right triangle. – Calculate the values of the six trig functions of the acute angles, given the measures of any 2 sides of a right triangle. Trig With Non-Acute Angles o Define the six circular trigonometric functions of an angle in standard position. – Relate the coordinates of a point on the terminal side of an angle to the position and dimensions of an appropriate reference triangle. – Calculate the values of the six trig functions of any angle (including non-acute), given the coordinates of a point on the terminal side of the angle. o Calculate the other five trigonometric function values of an angle, given one trigonometric function value and the quadrant in which the angle lies,. – Relate the definitions of the six trig functions to the position and dimensions of an appropriate reference triangle. Special Angles o Develop the unit circle, using both degrees and radians, recognizing and using the various patterns embedded in the circle. – Know/list (memorize) the measures of the special (and quadrantal) angles in both degrees and radians. – Know/list (memorize) the trigonometric function values of the special (and quadrantal) angles and their related angles in both degrees and radians. – Solve problems using the circular function definitions and the properties of the unit circle. o Recognize the connections between the coordinates of points on a unit circle and – coordinate geometry; – cosine and sine values; and – lengths of sides of special right triangles (30°-60°-90° and 45°-45°-90°). Inverses of the Special Angles o Calculate the measures of angles when their trigonometric function values are given. – Calculate the trigonometric function values of any angle in either degrees or radians. T.9 The student will identify, create, and solve practical problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines. T.4 The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function. Applications Using Calculators in Trig Solve Right Triangles o Solve a right triangle. – Calculate all three unknown measurements of any right triangle, given the measures of at least one side and one other dimension (angle or side). – Use the Pythagorean Theorem, trig functions, and inverse trig functions to calculate the unknown quantities. – Round calculated values to the appropriate number of significant digits, based on the precision of the given measurements. – Sketch diagrams of problems and correctly position values and variables (angles opposite appropriate sides). Applications o Solve practical problems (applications) involving triangles. – Sketch diagrams from word problems, placing values and variables in appropriate positions; ensure a correctly place right angle to allow solution of the problem using right triangle trig techniques. – Correctly place horizon lines to establish angles of elevation and/or depression for vertical application problems. – Correctly establish the right angle for horizontal application problems. Law of Sines, Ambiguous Case, Law of Cosines o Solve oblique (non-right) triangles, using combinations of the trigonometric functions, Pythagorean Theorem, Law of Sines, Law of Cosines, and the Angle Sum Theorem. – Identify the most efficient solution technique to be used with a given problem, based on the information (angle and side measures) given. – Law of Sines: Recogize as an extension of the definitions of the trig functions (SOHCAH-TOA) for oblique triangles, Requires a pair of opposites: angle and side Use when given AAS or ASA (with the Angle Sum Theorem) – Ambiguous Case of Law of Sines: Use when given SSA. Determine how many triangles exist for the given starting information. – Law of Cosines: Recogize as an extension of the PythagoreanTheorem for oblique triangles, Finds an opposite: angle or side Use when given SAS or SSS information. Recognize when and how to use the Law of Sines to determine follow-on values without falling into the Ambiguous Case. – Pythagorean Theoren and Angle Sum Theorem: Use to calculate required starting values for the 2 Laws, Use to find the measure of the 3rd angle of a triangle. o Calculate the area of any triangle. – Use the Laws of Sines and/or Cosines to calculate needed side lengths. o Calculate the altitude of a triangle from a known base. T.6 The student, given one of the six trigonometric functions in standard form [e.g., y = A sin (Bx + C) + D, where A, B, C, and D are real numbers], will a) state the domain and the range of the function; b) determine the amplitude, period, phase shift, and vertical shift; and c) sketch the graph of the function by using transformations for at least a oneperiod interval. Graphing Trig Functions Graphing Sine and Cosine (Amplitude and Period) Translating Sine and Cosine Graphs Equations from Graphs Cosecant and Secant Graphs Tan and Cot Graphs o Graph sin and cos curves. – Use the special angles in a table of values to craft graphs of the parent sine and cosine functions. – Expand the table of values (rows) to create graphs of transformed functions. – Recognize and identify the key points on the sine and cosine graphs; locations of : zeros (crossing points on the shifted xaxis) local maximum and minimum points, based on amplitude from the shifted xaxis. – From an equation in standard form, sketch a graph using the key-points method. o Graph sec, csc, tan, cot curves. – Use the special angles in a table of values to craft graphs of the parent functions. – Expand the table of values (rows) to create graphs of transformed functions. – Recognize and identify the key points on the graphs; locations of : zeros (crossing points on the shifted xaxis) local maximum and minimum points, based on amplitude from the shifted xaxis. Asymptotes Amplitude check points (tan and cot) – From an equation in standard form, sketch a graph using the key-points method. o Write the equation of the trig function, given its graph. o From the graph, determine critical information, including shifted origin, phase and vertical shifts, amplitude, and period. T.5 T.1 The student will verify basic trigonometric identities and make substitutions using the basic identities. The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles. Trigonometric Identities Fundamental Trig Identities Proving Trig Identities Sum and Difference Identities Double-Angle Identities Half-Angle Identities o Use the fundamental identities to find the values of the other 5 trig functions, given the value of one function. – Express one function in terms of another.. – Recognize the connection and equivalence between this use of the fundamental identities and the right triangle trig work done in Unit 1 using the definitions of the 6 trig functions along with the Pythagorean Theorem. o Write equivalent expressions. – Use the fundamental identities to rewrite an expression in terms of only sine and cosine. – Apply other standard algebra tools, including factoring. o Simplify trig expressions. – Use the fundamental identities to combine terms in an expression, with the goal of achieving a single term with a single trig function in the numerator. o Verify trigonmetric identities. – Prove that a trig equation is valid (true) for all meaningful values of (for every value in their domains). – Given an trig equation, use the trig identities so simplify each side of the equation independently to the point where both sides of the equation are identical. o Calculate exact values for the sine and cosine of the sum or difference of two angles. – Rewrite a given angle as the sum or difference of two special angles. – Use the sum and difference identities, along with the special angles, to calculate exact values. – Given a single trig function value of each of two angles, use right triangle trig tools (from Unit 1) to determine the sine and cosine of the angles, then use the appropriate identities to calculate the sine and cosine of the sum or difference of the given angles. o Calculate exact values for the sine and cosine of twice or half of a given angle. – Recognize the double-angle identity as a simplification of the sum identities, when the two angles are the same. As such, the sum identity can always be used in its place. – Recognize the half-angle identity as a simplification of the cosine sum identity, when the two angles are the same. As such, the cosine sum identity can always be used in its place when worked “backwards”. – Given a single trig function value of an angle, use right triangle trig tools (from Unit 1) to determine the sine and cosine of the angle, then use the appropriate identities to calculate the desired sum. G.5, G.6 Unit Objectives: The Learner Will: given the lengths of three segments, determine whether a triangle could be formed. arrange the angles of a triangle in order from smallest to largest when given the lengths of the sides. Inverse Trig Functions and Trig Equations Inverse Trig Functions Solving Trig Equations o Find the measures of angles in right triangles. – For the special angles, determine the angle measure when given a trig function ratio and quadrant for a point on the unit circle. Use coterminal angle concepts to arrange the sides of a triangle in order from smallest to largest when given the measures of the angles. given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie. determine all possible values for the angle. – Use the inverse trig functions to find angle measures when either: The sides of a right triangle are know, or When a trig function value is known, along with its quadrant. Determine all possible values for the angle. – Calculate the Principal Value for an inverse trig function. Sketch and identify the graphs of the inverse of each trig function. Determine the domain and range limitations of each in order for the result to remain a function. Use the range restrictions to calculate the Principal Value (also sometimes refered to as the Calculator Value or Function Value) Convert the Principal Value into an appropriate value within a given desired range. o Solve trigonometric equations with: – With either: Infinite solutions Restriced rangesolutions. – Using all trig and algebra tools learned so far, including: Factoring, Identities.