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SFSD Pre-Calculus Pacing Guide (created 2014-2015) Semester 2 Unit 6: Trigonometric Functions Notes: Total Suggested Number of Days for the Unit: 25 days Topic Unit Circle 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. Converting between degrees and radians Derive trigonometri c ratios on the unit circle using special right triangle 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 right triangles to determine geometrically I can… • Define radian, define degree, explain the need for radian measure. ● Define and use Initial side, terminal side, vertex, standard position, positive angle, negative angles, coterminal, central angles. ● Sketch angles in standard position. ● Find coterminal angles ● Find reference angle ● Convert angles from radians to degrees ● Convert angles from degrees to radians ● Construct the unit circle. ● Evaluate trig ratios on the unit circle ● Explain symmetry (odd and even) of the six trig functions using the unit circle. Resources Time Frame ● Section 4.1 ● Section 4.2 1-2 days ● Section 4.1 1 days ● Section 4.2 2-3 days 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, cosine, and tangent for π – x, π + x, and 2 π-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 function. Determine trig ratios off the unit circle given either a point or a trig ratio. 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. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. F.TF.8 Prove the Pythagorean identity sin^2(theta)+cos^2(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. (+) F.TF.3 Use special right 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, cosine, and tangent for π – x, π + x, and 2 π-x in terms of their values for x, where x is any real number. Graph Sine and Cosine Functions F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Graphs of Other Trigonometric F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline ● Evaluate trig ratios off the unit circle using triangles. ● Evaluate trig rations using the Pythagorean identity. ● Use the relationship between sine and cosine, secant and cosecant, and tangent and cotangent to find the trigonometric ratios of complementary angles. ● Section 4.3 ● Section 4.4 3-4 days ● Graph sine and cosine functions and find amplitude, frequency, midline, period, and phase shift. ● Write a sine or cosine function using given information. ● Use a trigonometric function to model data. ● Graph tangent, cotangent, secant, and cosecant functions. ● Find zeros and asymptotes of ● Section 4.5 3-4 days ● Section 4.6 2-3 days Functions tangent, cotangent, secant, and cosecant graphs. Inverse Trigonometric Functions (+) 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. Application of Right Triangles G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ ● Graph an arcsine, arccosine, and ● Section 4.7 arctangent function by limiting the domain. ● Evaluate an inverse trigonometric within the correct domain. ● Evaluate compositions of trigonometric functions. ● Use trigonometry to solve ● Section 4.8 problems involving right triangle. 3-4 days 2-3 days (+) 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. * Semester 2 Unit 7: Analytic Trigonometry Notes: Total Suggested Number of Days for the Unit: 25 days Topic Trigonometric Expressions and Identities Verifying Trigonometric Identities Standards I can… ● Factor a trigonometric expression ● Simplify trigonometric expressions. ● Rewrite fractional trigonometric expressions by applying a conjugate expression ● Verify a trigonometric identity. Resources Time Frame ● Section 5.1 4-5 days ● Section 5.2 4-5 days Solving Trigonometric Equations (+) F.TF. 7 Use inverse functions to solve trigonometric ● Solve trigonometric equations. equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. * ● Section 5.3 4-5 days Sum and Difference and Multiple Angle Identities (+) F.TF. 9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. ● Use sum and difference formulas ● Section 5.4 to evaluate sine, cosine, and ● Section 5.5 tangent expressions. ● Solve trigonometric equations using double angle formulas. 4-5 days Semester 2 Unit 8: Vectors Notes: Total Suggested Number of Days for the Unit: 18 days Topic Standards I can… Law of Sines and Cosines (+) G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems. ● Derive the law of sines and cosines. ● Use the law of sines and cosines to find missing parts of a triangle. ● Use the law of sines and cosines in application 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 (survey problems, resultant forces, etc.) Area of Triangles (+) G.SRT.9 Derive the formula A = ½ ab sin © for the area of a ● Find the area of triangles. triangle by drawing an auxiliary line from a vertex perpendicular to Resources Time Frame ● Section 6.1 ● Section 6.2 3-4 days ● Section 6.1 ● Section 6.2 1-2 days the opposite side. Vectors (+) N.VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes. (e.g., v, |v|,||v||,v) (+) N.VM.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. (+) N.VM.3 Solve problems involving velocity and other quantities that can be represented by vectors. (+) N.VM.4 Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v – w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. (+) N.VM.5 Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy)=(cvx,cvy). b. Compute the magnitude of a scalar multiple cv using ||cv||=|c|v. Compute the direction of cv knowing that when |c|v≠0, the direction of cv is either along v (for c > 0) or against v (for c < 0). (+) N.VM.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another ● Find the magnitude and ● Section 6.3 direction of a vector. ● Find the components of a vector given the initial and terminal point. ● Add and subtract vectors and multiply by a scalar. ● Understand graphically how to add and subtract vectors and multiply by a scalar. ● Use vectors to solve problems involving velocity and force. ● Find a transformation of a vector using matrices. 5-6 days vector. Work with matrices as transformations of vectors. Complex Numbers (+) N.CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers (+) N.CN.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. (+)N.CN. 5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1+√3i)3=8 because (–1+√3i) has modulus 2 and argument 120o. (+)N.CN.6 Calculate the distance between numbers in the complex plain as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. ● Write complex number in polar and rectangular form. ● Graph complex number in rectangular and polar form and show how they represent the same number. ● Add, subtract, multiply and conjugate a complex number. ● Find a power of a complex number. ● Find the distance and midpoint between numbers in the complex plane. ● Section 6.5 ● Section 9.5 3-4 days Semester 2 Unit 9: Polar Notes: Total Suggested Number of Days for the Unit: 10 days Topic Polar Coordinates Standards I can… ● Plot polar coordinates. ● Find multiple representations of points. ● Convert from rectangular to polar coordinates and polar to rectangular coordinates. Resources ● Section 9.5 (Consider teaching with complex numbers in section 6.5) Time Frame 2-3 days ● Convert rectangular to polar equations and polar to rectangular equations. ● Plot points to graph a polar equation. ● Graph polar equations using technology. ● Describe symmetry of polar graphs. ● Graph parametric equations by hand and using technology. ● Write parametric equations to model a horizontal and vertical component (projectile motion) ● Eliminate the parameter of parametric equations. Polar Equations Parametric Equations ● Section 9.6 2-3 days ● Section 9.4 2-3 days Semester 2 Unit 11: Limits and Derivatives Notes: Total Suggested Number of Days for the Unit: 7 days Topic Find Limits Graphically Find Limits Algebraically Standards I can… ● Use a table to estimate a limit. ● Use a graph to estimate a limit. ● Explain why a limit exists and does not exist. ● Find one sided limits. ● Evaluate a limit using substitution. ● Evaluate limits to infinity. ● Find one sided limits. Resources Time Frame ● Section 11.1 ● Section 11.2 2-3 days ● Section 11.1 ● Section 11.2 3-4 days