Pre-Calculus H Unit 2: Trigonometry Functions

Transcription

1 Pre-Calculus H Unit 2: Trigonometry Functions Unit #: APSDO Duration: 7.0 Week(s) Date(s): Team: Melinda Litke (Author), Tracy Andreana, Sally degozzaldi, Jennifer Greene, Jeanine LaBrosse, Jaclyn Lawlor, Melinda Litke, Ben Lukowicz, Jennifer Miller, Matthew Mooney, James Murray, Marlaina Napoli, Andrew Riddle, Steven Rivoira Grades: 10, 11, 12 Subjects: Mathematics Unit Focus In this unit, students will learn how to evaluate and graph the trigonometric functions, their inverses, and their reciprocals. They will be introduced to radian measures and the definitions of trigonometric ratios on the unit circle. Students will learn how to find trigonometric ratios of an acute angle by drawing a right triangle and of any angle by using reference angles in the unit circle. They will also learn how to graph trigonometric functions and identify the basic characteristics of those functions. Finally, students use trigonometric ratios to solve problems in a variety of contexts. Summative assessments may include projects, labs and test. Primary instructional materials for this unit include Pre- Calculus with Limits, Larson, Hostetler, and Edwards, Stage 1: Desired Results - Key Understandings Common Core Mathematics: 11 Established Goals Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. CCSS.MATH.CONTENT.HSF.TF.A.1 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Transfer T1 (T50) Based on an understanding of any problem, initiate a plan, execute it and evaluate the reasonableness of the solution. T2 (T53) Articulate how mathematical concepts relate to one another in the context of a problem or in the theoretical sense. T3 (T51) Examine alternate methods to accurately and efficiently solve problems. T4 (T52) Use appropriate tools strategically to deepen understanding of mathematical concepts. T5 (T32) Apply appropriate formulas to determine the unknown. T6 (T20) Compose and decompose numbers to establish relationships, perform operations, and solve problems. T7 (T23) Use functions or equations to model relationships among quantities. T8 (T24) Classify, interpret, and compare functions or equations. Page 1 of 5

2 CCSS.MATH.CONTENT.HSF.TF.A.2 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for x, pi + x, and 2pi - x in terms of their values for x, where x is any real number. CCSS.MATH.CONTENT.HSF.TF.A.3 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. CCSS.MATH.CONTENT.HSF.TF.A.4 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. CCSS.MATH.CONTENT.HSF.TF.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. CCSS.MATH.CONTENT.HSG.C.B.5 Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. CCSS.MATH.CONTENT.HSF.IF.C.7.E Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. CCSS.MATH.CONTENT.HSG.SRT.C.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. CCSS.MATH.CONTENT.HSF.TF.B.6 Explain and use the relationship between Understandings U1 (U408) Trigonometry is based on the relationship between sides and the angles in any triangle. U2 (U210) Trigonometric functions can be composed and decomposed to model a cyclical pattern and to solve problems. U3 (U207) Recognition of predictable mathematical patterns supports the analysis of functional relationships and the prediction of data. U4 (U510) Every problem is a member of a category of problems that has a similar structure and set of characteristics. U5 (U560) Patterns and structures are characterized by consistent relationships. U6 (U561) Recognition of patterns and structures fosters efficiency in solving problems. Meaning Acquisition of Knowledge and Skill Knowledge Essential Questions Q1 (Q406) What is the theorem/formula necessary to solve this problem? (Gr. 5-12) Q2 (Q409) How do some values in a triangle determine others? Q3 (Q208) What function best models the data? How do its characteristics help me make predictions? (Gr. 8-12) Q4 (Q200) What rule or pattern can help me simplify the expression or solve this problem? Q5 (Q205) How can I represent this relationship as a function or equation? (Gr. 6-12) Q6 (Q511) What characteristics/attributes define this type of problem? Q7 (Q560) What is the pattern/structure in this problem? Q8 (Q572) How does understanding the pattern/structure help me solve the problem? S1 Skills Convert between degrees and radians S2 Understand the relationship between degrees and radians S3 Understand the conversion between angles from degrees, minutes, seconds and vice versa S4 Page 2 of 5

3 the sine and cosine of complementary angles. CCSS.MATH.CONTENT.HSG.SRT.C.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. CCSS.MATH.CONTENT.HSF.TF.B.7 Prove the Pythagorean identity sin2(theta) + cos2(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle. CCSS.MATH.CONTENT.HSF.TF.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. CCSS.MATH.CONTENT.HSG.SRT.C.8 Look for and express regularity in repeated reasoning. CCSS.MATH.MP.8 Look for and make use of structure. CCSS.MATH.MP.7 Reason abstractly and quantitatively. CCSS.MATH.MP.2 Understand angles in standard position, reference angles, and coterminal angles S5 Solve problems involving linear and angular velocity S6 Understand the patterns of special right triangles and how they are related to the unit circle S7 Finding a trigonometric ratio of any acute angle by drawing a right triangle or using reference angles in the unit circle S8 Simplify trigonometric expression using reciprocal identities, quotient identities, and Pythagorean identities S9 Understand the relationship between the Pythagorean Theorem, equation of a circle and the Pythagorean identities S10 Understand the relationship between the unit circle and the graph of a trigonometric function S11 Use transformations to graph trigonometric functions from its parent functions S12 Identify key features of a trigonometric Page 3 of 5

4 function including periods, phase shift, amplitude, vertical shift, domain, range, intercepts, relative maximums and minimums, asymptotes, and limits S13 Understand the relationship between a trigonometric function and its reciprocal S14 Understand that the reciprocal of an x- intercept is a vertical asympote (undefined value) and vice versa S15 Write the equation of a trigonometric function given a graph, model, or set of data S16 Understand that the midline of trigonometric function is the average distance between the relative maximum and relative minimum S17 Using sinusoidal regression equations to make predictions and explain using real world connections S18 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed S19 Understand the difference between a trigonometric inverse and reciprocal S20 Page 4 of 5

5 Simplify expressions involving trigonometric and inverse trigonometric ratios S21 Understand that the amplitude is the distance between the relative maximum or minimum to the midline Stage 3: Learning Plan Coding Code Description of Learning Activity Page 5 of 5