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KUD Organizer Course: Analysis Trig Unit: Basic Trigonometry Approximate Days: 25 days STANDARDS: F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F-TF.2. 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. F-TF.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number. F-TF.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 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(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. F-TF.9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. UNDERSTAND: Understand trigonometric functions as they apply to the unit circle and all triangles. KNOW: Certain congruence criterion is required to use the Law of Sines. DO: Convert angle units to and from degree and radian measure. Identify coterminal, quadrantal, reference angles. How to use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6 Compare and contrast right and non-right triangles. Classify the law of sines and the law of cosines to explain their proofs. 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. By similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent. Use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number. 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. Demonstrate that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent. The relationship between the sine and cosine of complementary angles. Use and explain the relationship between the sine and cosine of complementary angles. Law of Cosines generalizes the Pythagorean Theorem. Use sine, cosine, tangent and the Pythagorean Theorem to solve right triangles in applied problems. Law of Sines and Cosines embody the triangle congruence criteria, in that three pieces of information are usually sufficient to completely solve a triangle; and yield two possible solutions in the ambiguous Explain the proofs of the Law of Sines and the Law of Cosines. Use 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). case illustrating that SSA is not congruence criterion. Radian measure of an angle is the length of the arc on the unit circle subtended by the angle. KEY VOCABULARY: Trigonometry, trigonometric functions, Pythagorean Theorem, unit circle, congruence postulates, corresponding parts of triangles, angle relationships, coterminal angles, quadrantal angles, reference angle, radians, degrees, Law of Sines, Law of Cosines