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Year: 2014-15 Teacher: CORE Trigonometry (SEM) Course: Trigonometry (SEM) Graphs and Functions ~ Standards Month: All Months In this unit we will explore how to find the distance between two points and the mid-point of a line segment. We will also identify the important part of circles and how they relate to their respective equations. Essential Questions Assessments Skills Content Lessons Resources G-GPE.7-Use coordinates to compute What is the distance between Quiz 1-1 - 1-2 9/1/2014 Use the distance formula Distance perimeters of polygons and areas of two points on the coordinate plan Quiz Special Right Use the midpont formula Midpoint triangles and rectangles, e.g., using the Triangles 9/15/2014 distance formula. Write the standard form of Circles and their G.2.1.1-Solve problems involving right the equation of a circle equations What is the midpoint between triangles. Graph a circle G-GPE.1-Derive the equation of a circle of two points on a coordinate plan? given center and radius using the Work with the general form of How can you find the missing Pythagorean Theorem; complete the the equation of a circle sides in a 45-45-90 triangle when square to find the center and radius of a given just one side? circle given by an equation. G-SRT.8-Use trigonometric ratios and the Pythagorean Theorem to solve right How can you find the missing triangles in applied problems. G-SRT.9-(+) Derive the formula A = 1/2 ab sides of a 30-60-90 triangle when sin(C) for the area of a triangle by drawing given just one side? an auxiliary line from a vertex perpendicular to the opposite side. Trigonometric Functions ~ Standards In this section we will the basics of trig. We will learn how to convert from degrees to radians and vice versa. We will learn the unit circle values and how to apply that information. We will work with transformations in order to graph different trig functions. Essential Questions Assessments Skills F-TF.1-Understand radian measure of an angle as the 1. To convert from Quiz 2-1 - 2-2 length of the arc on the unit circle subtended by the degree to radians what 10/1/2014 angle. is the conversion Quiz 2-3 - 2-4 F-TF.2-Explain how the unit circle in the coordinate factor? 10/14/2014 plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of Quiz 2-6 2. In a 30-60-90 right angles traversed counterclockwise around the unit 10/21/2014 triangle, the circle. hypotenuse is what Chapter 2 Test F-TF.3-(+) Use special triangles to determine times the shortest side? 10/31/2014 geometrically the values of sine, cosine, tangent for ?/3, ?/4 and ?/6, and use the unit circle to express the 3. What does the values of sine, cosines, and tangent for x, ? + x, and 2? coefficient in front of the – x in terms of their values for x, where x is any real trig function affect? number. F-TF.4-(+) Use the unit circle to explain symmetry (odd 4. What does the and even) and periodicity of trigonometric functions. coefficient in front of the F-TF.5-Choose trigonometric functions to model x in a trig function periodic phenomena with specified amplitude, affect? frequency, and midline. 1. Convert between degrees and degrees, minutes, and seconds. 2. Convert between degrees and radians. 3. Calculate the arc length and area of a sector. 4. Review special right triangles. 5. Calculate unit circle values. 6. Determine the domain and range of trig functions. 7. Calculate trig values when in different quadrants. 8. Using trig identities, calculate values. 9. Graph using transformations. 10. Calculate the amplitude, fundamental period, and phase shifts of sinusoidal curves. 11. Write the equation of the graph of a sinusoidal curve. 12. Sketch the remaining trig functions and calculate their values. Content 1. radians 2. conversion factors 3. unit circle values 4. trig values 5. graphs of six trig functions 6. transformations 7. sinusoidal curves 8. domain and range 9. equations of graphs Lessons Resources Trigonometry, Seventh EditionLarson and Hostetler, Houghton Mifflin F-TF.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. F-TF.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. F-TF.8-Prove the Pythagorean identity sin2(?) + cos2(?) = 1 and use it to calculate trigonometric ratios. F-TF.9-(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. G-C.1-Prove that all circles are similar. G-SRT.7-Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.8-Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 5. How do I determine the fundamental period from the equation? 6. All the functions have a fundamental period of ___, except for the ____ and _____ function, which have fundamental periods of ____. Law of Sines and Law of Cosines Standards Essential Assessments Skills Questions F-TF.1-Understand radian measure of an angle as the length of 1. Give one Quiz 4-1 the arc on the unit circle subtended by the angle. of the law of 12/1/2014 F-TF.2-Explain how the unit circle in the coordinate plane sine formulas Quiz 4-2 enables the extension of trigonometric functions to all real for area. 12/8/2014 numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 2. Which law F-TF.3-(+) Use special triangles to determine geometrically the has an values of sine, cosine, tangent for ?/3, ?/4 and ?/6, and use the ambiguous unit circle to express the values of sine, cosines, and tangent for case? x, ? + x, and 2? – x in terms of their values for x, where x is any real number. 3. What do we F-TF.4-(+) Use the unit circle to explain symmetry (odd and mean by an even) and periodicity of trigonometric functions. ambiguous F-TF.5-Choose trigonometric functions to model periodic case? phenomena with specified amplitude, frequency, and midline. F-TF.6-(+) Understand that restricting a trigonometric function to 4. In order to a domain on which it is always increasing or always decreasing use the law of allows its inverse to be constructed. cosine we trace F-TF.7-(+) Use inverse functions to solve trigonometric what from the equations that arise in modeling contexts; evaluate the solutions diagram? using technology, and interpret them in terms of the context. F-TF.8-Prove the Pythagorean identity sin2(?) + cos2(?) = 1 and 5. In order to use it to calculate trigonometric ratios. use the law of F-TF.9-(+) Prove the addition and subtraction formulas for sine, sine to find cosine, and tangent and use them to solve problems. area, what do G-SRT.7-Explain and use the relationship between the sine and we trace in the cosine of complementary angles. diagram? G-SRT.8-Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G-SRT.10-(+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.11-(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces). 1. Learning and applying the law of sines Content 1. Law of sines 2. Learning and applying the law of cosines 2. Law of cosines 3. Calculating the area of triangles 4. Drawing, using and applying vectors 3. vectors Lessons Resources Law of sines Trigonometry, Seventh EditionLarson and Hostetler, Brooks and Cole Analytic Trigonometry Standards Essential Assessments Questions F-TF.1-Understand radian measure of an angle as the length 1 cos x/2 = of the arc on the unit circle subtended by the angle. F-TF.2-Explain how the unit circle in the coordinate plane 2. cos( a -b)= enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed 3. What does counterclockwise around the unit circle. the cos (π/2 F-TF.3-(+) Use special triangles to determine geometrically x)= the values of sine, cosine, tangent for ?/3, ?/4 and ?/6, and use the unit circle to express the values of sine, cosines, and 4. If I'm using tangent for x, ? + x, and 2? – x in terms of their values for x, cos 4x where x is any real number. = , explain F-TF.4-(+) Use the unit circle to explain symmetry (odd and how to find all even) and periodicity of trigonometric functions. possible roots F-TF.5-Choose trigonometric functions to model periodic for this phenomena with specified amplitude, frequency, and midline. equation. F-TF.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. F-TF.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. F-TF.8-Prove the Pythagorean identity sin2(?) + cos2(?) = 1 and use it to calculate trigonometric ratios. F-TF.9-(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. G-SRT.4-Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT.8-Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G-SRT.10-(+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.11-(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Analytic Trig test 12/1/2014 Skills 1. Evaluating trig identities. 2. Verifying trig identities Quiz on verifications and simplifications 3. Solving trig equations. 12/1/2014 Trig equation quiz 12/1/2014 angle measure quiz 12/1/2014 4. Learning and applying sum and difference formulas. Content Lessons Resources 1. Verifications 2. Simplifications 3. Trig equations 4. sum and difference formulas 5. Learning and applying multiple 5. multiple angle formulas angle and product to sum formulas. 6. product to sum formulas Difference Trigonometry, between a Seventh Editionverification Larson and and a Hostetler, simplification Brooks and Cole