Trigonometry PUHSD Curriculum

Transcription

1 Unit 1: Right Triangle Trigonometry The relationship between the sides and angles of right triangles leads to the exploration of trigonometric functions. By using the special relationships of the 30º-60º-90º triangle and the 45º-45º-90º triangle, the meaning of trigonometric functions becomes clear, and that knowledge leads to understanding how that applies to real-world situations. How are proportions set up for similar polygons? What are the six trigonometric functions for the acute angles in a right triangle? Why are the trigonometric ratios in similar triangles equal? What are the ratios of the sides of a 45º-45º-90º triangle and a 30º-60º-90º triangle? How is trigonometry used to solve right triangles, including real-world applications? 7.RP.2 G-SRT.6 F-TF.3 G-SRT.8 Recognize and represent proportional relationships between quantities. Evaluate the six trigonometric functions of the acute angles of a right triangle. 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. Solve a 45º-45º-90º triangle or a 30º-60º-90º given the length of one side. Use special triangles to determine geometrically the values of sine, cosine, and tangent for 30º, 45º, and 60º. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. I can use similar figures to write a proportion. I can write a proportion for similar triangles. Given an acute angle in a right triangle, I can identify the opposite side, the adjacent side and the hypotenuse. I can evaluate the six trigonometric functions of the acute of a right triangle. I can explain why two angles of equal measure in different size triangles have the same trigonometric ratios. I can draw and label a 45º-45º-90º special triangle. I can draw and label a 30º-60º-90º special triangle. I can solve a special triangle. I can state the values of the trigonometric functions of the special angles. I can find the missing sides of a right triangle using the trigonometric ratios. I can find the acute angles in a right triangle using an inverse trigonometric function. I can solve a real-world problem that can be Use a calculator to find the sine, cosine, and tangent of an angle. Use reciprocals on a calculator to find the cosecant, secant, and cotangent of an angle. Use technology to investigate the special triangles. (Cabri Jr., Geo Sketchpad, TI-Nspire, Excel) Use SIN -1, COS -1, and TAN -1 to approximate values of inverse trigonometric functions. proportion acute angle right triangle trigonometry trigonometric ratio opposite adjacent hypotenuse trigonometric function sine Page 1 of 16

2 modeled with right triangle trigonometry. cosine tangent secant cosecant cotangent special triangles Pythagorean Theorem Page 2 of 16

3 Unit 2: The Unit Circle Using special right triangles builds understanding of the relationships in the unit circle. The relationships between the four quadrants and the six trigonometric functions make the solving of equations clear. The equation for the circumference of a circle makes the relationship between degrees and radians clear. How are positive and negative angles of all sizes represented on a unit circle? How are the x and y coordinates of a point related to the angles and their trigonometric functions? How is a reference angle used to find trigonometric functions in all quadrants? What is radian measure and how do we convert between degrees and radians? How are odd, even, and periodicity of trigonometric functions explained using the unit circle? How is arc length and sector area found when the radius and central angle (in degrees or radians) are given? F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as angles (expressed in degrees) traversed counterclockwise and clockwise around the unit circle. F-TF.3 F-TF.1 F-TF.2 Find the trigonometric functions of an angle in any quadrant. Use special triangles to determine geometrically the values of sine, cosine, tangent for 30º, 45º, and 60º and use the unit circle to express the values of sine, cosine, and tangent for 180º x, 180º + x and 360º x in terms of their values for x, where x is 30º, 45º, or 60º. Solve basic trigonometric equations involving special triangles, such as 2sin 1 0, where 0º < θ < 360º. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as angles (expressed in I can draw an angle in standard position (positive and negative). I can determine the measures of angles (positive and negative) which are coterminal with a given angle. I can use trigonometric functions to find the coordinates of a point on the terminal side of an angle in standard position. I can determine the measure of the angle, given the coordinates of a point on the terminal side. I can define the trigonometric functions in terms of x, y, and r. I can evaluate the trigonometric functions of 0º, 90º, 180º, 270º, and 360º. I can label the unit circle in degrees. I can determine the reference angle for an angle in any quadrant. I can determine the sign of the trigonometric functions in each quadrant. I can evaluate the six trigonometric functions of any angle. I can state the values of the trigonometric functions of the special angles in degrees. Use a calculator to convert degree measures to radian measures. Use a calculator to convert radian measures to degree measures Evaluate all six trigonometric functions of angles in radian measure using the calculator in radian mode and in degree measure using the calculator in degree mode. Determine when it is appropriate to use radian or degree mode on a calculator. coordinate plane quadrants unit circle clockwise counterclockwise reference angles Page 3 of 16

4 F-TF.3 F-TF.4 G-C.5 radians) traversed counterclockwise and clockwise around the unit circle. Use special triangles to determine geometrically the values of sine, cosine, tangent for,, and and use the unit circle to express the values of sine, cosine, and tangent for x, x, and 2 x in terms of their values for x, where x is,, or Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 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. I can evaluate the six trigonometric functions of an angle in any quadrant whose reference angle is 30º, 45º, or 60º. I can use the unit circle to evaluate the trigonometric functions of the special angles in any quadrant. I can solve basic trigonometric equations for missing angles, remembering to find all possible solutions. I can define the meaning of radian measure. I can convert between degree and radian measure with and without technology. I can represent radian measure in terms of π. I can use decimal representation for radian measures. I can evaluate the six trigonometric functions of angles in radian measure. I can draw an angle in standard position (positive and negative). I can determine the measures of angles (positive and negative) which are coterminal with a given angle. I can use trigonometric functions to find the coordinates of a point on the terminal side of an angle in standard position. I can determine the measure of the angle, given the coordinates of a point on the terminal side. I can define the trigonometric functions in terms of x, y, and r. I can evaluate the trigonometric functions of 0, 3,,, and I can label the unit circle in degrees. I can label the unit circle in radians. quadrantal angles radian arc arc length circumference subtended central angle symmetry even/odd function period periodicity constant of proportionality sector Page 4 of 16

5 I can state the values of the trigonometric functions of the special angles in radian measure without converting the angle measures to degrees first. I can evaluate the six trigonometric functions of an angle in any quadrant whose reference angle is,, or I can use the unit circle to evaluate the trigonometric functions of the special angles in any quadrant. I can determine the relationship between the sin (cos, tan) of θ and the sin (cos, tan) of θ. I can use the periodicity of trigonometric functions to label the cos and sin of the angles in the unit circle. I can find the length of an arc given the radius and the measure of the central angle in degrees or in radians. I can find the area of a sector given the measure of the central angle in degrees or in radians. Page 5 of 16

6 Unit 3: Graphs of Trigonometric Functions The unit circle and relationships on it can be and are translated to the coordinate plane. The movement and shifts of all graphs can be shown in trigonometric functions. What are amplitude, midline, frequency and period in relation to graphs of trigonometric functions? Which of the six trigonometric functions have asymptotes and where do they occur? What do the key features (max, min, increasing, decreasing, etc.) tell us about trigonometric graphs and what do they represent in real-world situations modeled by trigonometric functions? How does changing the parameters affect the graphs of trigonometric functions? (shifts, stretches, ) What real-world situations are modeled by trigonometric functions? F-IF.7e Graph trigonometric functions, showing period, midline, and amplitude. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; end behavior; and periodicity. F-BF.3 Identify the effect on the graph by replacing f( x ) with f ( x) k, kf ( x ), f ( kx ), and f ( x k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic ACT Quality Core F.3.f F-TF.5 expressions for them. Write the equations of sine and cosine functions given the amplitude, period, phase shift, and vertical translation. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. I can identify the midline, amplitude, and period of a sine and cosine function. I can sketch the graphs of the sine and cosine functions. I can identify the midline, period, and asymptotes of a tangent function. I can sketch the graph of a tangent function. I can sketch the graphs of the secant, cosecant, and cotangent functions. I can find the domain and range of a trigonometric function. I can find the intercepts of a trigonometric function. I can identify intervals where a trigonometric function is increasing, decreasing, positive, or negative. I can find relative maximums and minimums of a trigonometric function. I can identify any symmetry in the graph of a trigonometric function. I can use vertical and horizontal Use amplitude, period, and frequency to determine an appropriate window for graphing a trigonometric function on a calculator. Graph a trigonometric function on a calculator in degree mode and in radian mode. Verify the solution to a system of trigonometric functions. Use the domain and range to set an appropriate window for graphing a trig function on a calculator. Use the TRACE and CALCulate features to find a specific value, the intercepts, and relative maxima and minima of a trig function. Use the TABLE feature of a calculator to identify key features of a trig function. Describe the period, domain, range and asymptotes of a trigonometric function graphed on a calculator. Use a graphing calculator to experiment with changing the Page 6 of 16

7 shifts to sketch graphs of trigonometric functions. I can use vertical and horizontal stretching to sketch graphs of trigonometric functions. I can use reflections to sketch graphs of trigonometric functions. I can classify the transformation by comparing two given graphs. I can determine whether a trigonometric function is even or odd. I can use the midline, amplitude, period, and phase shift to write the equation of a graph of a sine and cosine function. I can write the equation of a sinusoid as a sine function and as a cosine function. I can apply the meaning of the key features of a trigonometric function to a real-world situation. I can solve real-world problems that can be modeled with trigonometric functions. parameters of an equation of a trigonometric function and determine their effect on the graph. Use a calculator to translate sine, cosine and tangent graphs. Use a calculator to illustrate how the graphs of sine, cosine and tangent are affected by translations. Graph a trigonometric model of a real-world problem and use the various features of the calculator to solve the problem. midline amplitude frequency phase shift vertical translation domain range intercepts asymptote interval increasing/decreasing maximum/minimum symmetry end behavior periodicity parameter even/odd function Page 7 of 16

8 A firm understanding of domain and range and the inverse of functions is applied to trigonometric functions. Unit 4: Inverse Trigonometric Functions Since the trigonometric functions are not one-to-one, how can the domain be restricted to graph the inverse functions? How are inverse trigonometric functions used to find angles in real-world problems? F-BF.4c Read values of an inverse function from a graph or a table, given that the function has an inverse. F-TF.6 F-BF.4b G-SRT.7 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. Verify by composition that one function is the inverse of another. Evaluate expressions containing inverse trigonometric functions. Explain and use the relationship between the sine and cosine (and other functions) of complementary angles. I can determine the coordinates of the points on an inverse trigonometric function from a table of values. I can explain why the domain of the sine, cosine and tangent functions must be restricted in order to have an inverse. I can determine the domain for the inverse sine, inverse cosine, and inverse tangent functions. I can graph the arcsin, arccos, and arctan functions. I can use composition of functions to solve problems such as arcsin(sin ) and sin(arcsin x ). I can evaluate expressions such as 1 tan arcsin 2. I can explain why the sin cos 2. I can explain why the sec csc 2. I can explain why the tan cot 2. Choose an appropriate window on a calculator to graph the arcsin, arccos, and arctan functions. Use the calculator to evaluate expressions such as tan arcsin inverse vertical line test horizontal line test one-to-one invertible/non-invertible composition arcsine arccosine arctangent sin 1 cos 1 tan 1 Page 8 of 16

9 Unit 5: Trigonometric Identities The basis of trigonometric identities comes from both the unit circle and the Pythagorean Theorem. Manipulation, substitution, and reduction can simplify complex trigonometric expressions. How can the Pythagorean, reciprocal, co-function, even/odd, and quotient trigonometric identities be developed from the unit circle? How can algebraic operations be used to simplify trigonometric expressions and verify trigonometric identities? How can trigonometric identities for sum, difference, double angle and half angle simplify evaluating expressions and solving equations? F-TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F-TF.8 ACT Quality Core F.3.i ACT Quality Core F.3.g ACT Quality Core F.3.h ACT Quality Core F.3.i 2 2 Prove the Pythagorean identity sin cos 1 and use it to find tan, sec, csc and cot. Apply the fundamental trigonometric identities to simplify and evaluate trigonometric expressions and prove trigonometric identities. Identify the sum and difference identities for the sine, cosine, and tangent functions; apply the identities to solve mathematical problems. Derive, identify, and apply double-angle and half-angle formulas to solve mathematical problems. Apply the fundamental trigonometric identities, the doubleangle and half-angle identities, and the sum and difference identities to simplify and evaluate trigonometric expressions and prove trigonometric identities. I can determine the relationship between the sin (cos, tan) of θ and the sin (cos, tan) of θ. I can use the unit circle to illustrate the 2 2 Pythagorean identity sin cos 1. I can use the identity 2 2 sin cos 1 to develop the other Pythagorean identities and the quotient identities. I can use the fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. I can verify trigonometric identities using the quotient, reciprocal, co-function, even/odd, and Pythagorean identities. I can state the identities for the sine and cosine of the sum and difference of two angles. I can use the sum and difference formulas to evaluate trigonometric functions without using a calculator. I can use the identities for the sine and cosine of sums and differences to develop the identities for the tangent of sums and Verify trigonometric identities by graphing. Verify the solution to a system of trigonometric functions. trigonometric identities quotient identities Pythagorean identities sum/difference formulas double-angle formulas half-angle formulas co-functions even/odd identities simplify evaluate verify solve Page 9 of 16

10 differences. I can use the identities for the sine and cosine of sums and differences to develop the double-angle and half-angle identities. I can use the sum and difference formulas to evaluate trigonometric functions, verify trigonometric identities, and solve trigonometric equations. I can use multiple-angle formulas to rewrite and evaluate trigonometric functions. I can use half-angle formulas to rewrite and evaluate trigonometric functions. Page 10 of 16

11 Unit 6: Trigonometric Equations The rules for solving all types of equations are integrated with the identities of trigonometry in order to solve trigonometric equations. The equations are solved using the understanding of inverse trigonometric functions in conjunction with the unit circle. How are algebraic operations used for solving trigonometric equations (including those in quadratic form)? What substitutions involving trigonometric identities need to be used for solving some trigonometric equations? How can we represent and solve real-world problems with trigonometric equations? F-TF.7 Use inverse functions to solve trigonometric equations. I can use inverse functions to solve trigonometric equations. A-REI.4b A-REI.10 F-TF.7 2 x 49 ), Solve quadratic equations by inspection (e.g., for taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Use sum/difference, multiple-angle and half-angle formulas to solve trigonometric equations. Solve advanced trigonometric equations. 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. I can use and apply the Pythagorean and quotient identities to simplify and solve trigonometric equations. I can solve trigonometric equations in quadratic form by taking square roots, by completing the square, by using the quadratic formula, and by factoring. I can solve trigonometric equations by graphing. I can use the sum and difference formulas to solve trigonometric equations. I can use multiple-angle formulas to solve trigonometric equations. I can use half-angle formulas to solve trigonometric equations. I can solve advanced trigonometric equations in quadratic form, including those which involve a substitution. I can solve advanced trigonometric equations involving multiple angles. I can solve real-world problems that can be modeled with trigonometric equations. Use SIN -1, COS -1, and TAN -1 to approximate values of inverse trigonometric functions. Use the zero or root feature to approximate x-intercepts of trigonometric functions. Use a graphing calculator to solve a trigonometric equation. Use a calculator to solve a trigonometric equation trigonometric equation quadratic completing the square quadratic formula factoring zeros of a function domain Page 11 of 16

12 Unit 7: Additional Topics in Trigonometry Trigonometry is expanded beyond the right triangle. The sides and angles of all triangles can be found, providing solutions to real-world problems. How is the area of a triangle found when two sides and the included angles are given? How are oblique triangles solved using the Law of Sines and the Law of Cosines? In real-world situations, such as navigation, surveying, etc., how can the Law of Sines of the Law of Cosines be used? G-SRT.9 1 I can derive the formula Derive the formula A absinc for the area of a triangle by 2 1 A absinc. drawing an auxiliary line from a vertex perpendicular to the 2 auxiliary line opposite side. I can find the area of a triangle perpendicular ACT Use various methods to find the area of a triangle (e.g., given the given two sides and the included oblique triangle Quality length of two sides and the included angle). angle. Law of Sines Core F.3.a I can use the Law of Sines to solve Law of Cosines G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve an oblique triangle. problems. I can use the Law of Cosines to 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). solve an oblique triangle. I can use the Law of Sines and Law of Cosines to solve real-world problems. Page 12 of 16

13 Unit 8: Vectors By combining the properties of linear equations, directionality, and trigonometry, vectors bring new dimensions to mathematics. What notation is used to represent vectors? Given the initial and terminal point on a vector, what is the component form? What is velocity and how are vectors used to represent it? How can operations on vectors (+, -, x, ) be performed graphically and algebraically? 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 N-VM.3 N-VM.4a N-VM.4b N-VM.4c N-VM.5a Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Solve problems involving velocity and other quantities that can be represented by vectors. 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. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 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. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v x, v y ) = (cv x, cv y ). N-VM. 5b 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). Apply the properties of vectors to finding angle measures. I can represent a vector as a directed line segment. I can use appropriate symbols for vectors and their magnitudes. I can represent a vector in component form. I can represent a vector in linear form. I can calculate the magnitude of a vector. I can use the Law of Sines and the Law of Cosines to solve real-world problems. I can solve real-world problems involving quantities, including velocity, force, and work, that can be represented by vectors. I can add vectors geometrically by using the parallelogram rule. I can add vectors in component form and in linear form. I can find the magnitude of a sum of two vectors. I can find the sum of the magnitudes of two vectors. I can show that the magnitude of a sum of two vectors is not the same as the sum of the magnitudes of two vectors. Use inverse trigonometric functions to find angle measures. vector magnitude directed line segment component form parallelogram rule initial terminal velocity resultant scalar Page 13 of 16

14 N-CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. I can represent a vector in magnitude and direction form. I can find the magnitude and the direction of the sum of two vectors. I can represent vector subtraction geometrically. I can subtract vectors in component form and in linear form. I can represent scalar multiplication geometrically. I can multiply a vector in component form and in linear form by a scalar. I can calculate the magnitude of a scalar multiple of a vector. I can determine the direction angle for a vector. I can find the dot product of two vectors and use the properties of the dot product. I can find the angle between two vectors. I can find the distance between two numbers in the complex plane. I can find the midpoint of segment joining two numbers in the complex plane. Page 14 of 16

15 With the use of imaginary numbers solutions are no longer restricted to the tangible world. Complex numbers are often converted into polar form to simplify the procedure of performing some mathematical operations. Unit 9: Complex Numbers How are complex numbers written in polar form? How can we perform computations with complex numbers in polar form? N-CN.1 Know there is a complex number i such that i 2 1, and every complex number has the form a + bi with a and b real. N-CN.2 N-CN.3 N-CN.4 N-CN.5 2 Use the relation i 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 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. Represent multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this 1 3i 8 representation for computation. For example, 3 because 1 3i has modulus 2 and argument 300. I can write any complex number in standard form. I can add, subtract, and multiply complex numbers in standard form. I can determine the conjugate of a complex number. I can divide complex numbers in standard form. I can find the absolute value of a complex number. I can graph a complex number in rectangular form. I can graph a complex number in polar form. I can convert from rectangular coordinates to polar coordinates. I can convert from polar coordinates to rectangular coordinates. I can multiply and divide complex numbers written in trigonometric form. I can use DeMoivre s Theorem to find powers of complex numbers. imaginary number complex number i standard form complex plane commutative property associative property distributive property conjugate modulus (magnitude) argument rectangular form polar form r cis θ De Moivre s Theorem Page 15 of 16

16 Enduring Understanding: Polar graphs add aesthetics to geometric designs. Unit 10: Polar Equations and Graphs Essential Question: Which parameters and which functions change the families of polar curves? Graph and analyze polar equations. I can plot points and find multiple representations of points in the polar coordinate system. I can convert points between rectangular and polar coordinates. I can convert equations between rectangular and polar form. I can graph polar equations by point plotting. I can use symmetry as a sketching aid for polar graphs. I can recognize special polar graphs. Use a calculator in POLar mode to graph polar equations. Use the table feature to verify points on a polar graph. polar equation families of polar curves Page 16 of 16