Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 4.7 Inverse Trig Functions Syllabus Objective: 4.2 – The student will sketch the graphs of the principal inverses of the six trigonometric functions. Recall: In order for a function to have an inverse function, it must be one-to-one (must pass both the horizontal and vertical line tests). Notation: The inverse of f x is labeled as f 1 x . y sin x Graph of f x sin x Domain: Range: In order for f x sin x to have an inverse function, we must restrict its domain to , . 2 2 Inverse of the Sine Function To graph the inverse of sine, reflect about the line y x . Domain of f 1 x sin 1 x : Notation: Inverse of Sine Range of f 1 x sin 1 x : f 1 x sin 1 x or y arcsin x (arcsine) Note: y sin1 x denotes the inverse of sine (arcsine). It is NOT the reciprocal of sine (cosecant). Page 1 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Evaluating the Inverse Sine Function Ex1: Find the exact values of the following. 1. arcsin 2 2 What value of x makes the equation sin x 2 true? 2 Note: The range of arcsine is restricted to , , so _____ is the only possible answer. 2 2 1 2. sin 3 What value of x makes the equation sin x 3 true? ____________________ 3. 2 sin 1 sin Taking the inverse sine of the sine function may result in the argument, but not 3 always. Sometimes it will be the reference angle in the appropriate quadrant. Inverse of the Cosine Function Graph of f x cos x Domain: Range: In order for f x cos x to have an inverse function, we must restrict its domain to 0, . To graph the inverse of cosine, reflect about the line y x . Domain of f 1 x cos1 x : Notation: Inverse of Cosine Range of f 1 x cos1 x : f 1 x cos1 x or y arccos x (arccosine) Note: y cos1 x denotes the inverse of cosine (arccosine). It is NOT the reciprocal of cosine (secant). Page 2 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Evaluating the Inverse Cosine Function Ex2: Find the exact values of the following. 1. 2 2 arccos true? What value of x makes the equation cos x 2 2 Note: The range of arccos is restricted to 0, , so ________ is the only possible answer. 2. 3 sin cos1 2 3. 11 cos1 cos 6 3 cos 1 2 , so sin 6 Inverse of the Tangent Function Graph of f x tan x Domain: Range: In order for f x tan x to have an inverse function, we must restrict its domain to , . 2 2 To graph the inverse of tangent, reflect about the line y x . Domain of f 1 x tan 1 x : Notation: Inverse of Tangent Range of f 1 x tan 1 x : f 1 x tan 1 x or y arctan x (arctangent) Note: y tan 1 x denotes the inverse of tangent (arctangent). It is NOT the reciprocal of tangent (cotangent). Page 3 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Evaluating the Inverse Tangent Function Ex3: Find the exact values of the following. 1. 3 sin tan 1 3 3 sin tan 1 sin________ ________ 3 3 Note: The range of arctangent is restricted to , , so ___ is the only possible answer for tan 1 . 3 2 2 2. cos tan 1 1 cos tan 1 1 cos______ ________ 3. arccos tan 3 arccos tan arccos______ No Solution, because _______ 3 You Try: Evaluate cos 1 cos . Be careful! 4 QOD: Explain how the domains of sine, cosine, and tangent must be restricted in order to create an inverse function for each. Page 4 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 4.7 Ex.1 Inverse Trig Functions Continued Find the values of x a) x = cos-1(cos x) b) x = cos(cos-1 x) Ex.2 Find the exact solution to the equation without a calculator: a) 6tan-1 x = π b) sin-1 (sin x) = 6π/7 Ex.3 Use the triangle to answer the questions. a) Find tan θ. b) Find tan-1 x. x θ 1 c) Find the hypotenuse as a function of x. d) Find sin (tan-1 (x)) as a ratio involving no trig f(x)s. e) Find sec (tan-1 (x)) as a ratio involving no trig f(x)s. f) If x < 0, then tan-1 x is a negative angle in the fourth quadrant. Verify d & e are still valid. Right Triangle Trigonometry and Inverse Trigonometric Functions: the trigonometric functions can be evaluated without having to find the angle Label the sides of the right triangle based upon the inverse trig function given Evaluate the length of the missing side (Pythagorean Theorem) Evaluate the trig function – be sure to choose the correct sign! Remember: the inverse trig functions are angles! We don’t actually need to find θ to solve. Always think of the restricted domain! Page 5 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities 1 Ex4: Evaluate cos arctan without a calculator. 5 θ Right Triangle Hypotenuse: 1 Let arctan . Since the range of arctangent is , , and the tangent is positive, must be in 5 2 2 Quadrant ______. Therefore, cosine is positive. So cos . Ex5: Find an algebraic expression equivalent to sin arccos 4x . θ Ex.6 Ex.7 Find an algebraic expression equivalent to: a) sin (cos-1 v) A 15 ft wide billboard is placed ly 10 ft from a road. A spotlight at the edge of the road is aimed at the sign. a) Express θ as a function of the distance (x) from 10 ft point A to the spotlight. A θ θ = tan-1 (25/x) – tan-1 (10/x) b) How far from point A should the spotlight be placed to maximize θ? x ≈ 15.81 feet Page 6 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 4.8 Trig Application Problems Syllabus Objective: 4.5 – The student will model real-world application problems involving graphs of trigonometric functions. Angle of Elevation: the angle through which the eye moves up from horizontal to look at something above Angle of Depression: the angle through which the eye moves down from horizontal to look at something below Angle of Elevation Angle of Depression Solving Application Problems with Trigonometry: Draw and label a diagram (Note: Diagrams shown are not drawn to scale.) Find a right triangle involved and write an equation using a trigonometric function Solve for the variable in the equation Note: Be sure your calculator is in the correct Mode (degrees/radians). Ex1: If you stand 12 feet from a statue, the angle of elevation to the top is 30°, and the angle of depression to the bottom is 15°. How tall is the statue? Height of the statue is approximately Ex2: Two boats lie in a straight line with the base of a cliff 21 meters above the water. The angles of depression are 53° to the nearest boat and 27° to the farthest boat. How far apart are the boats? Distance between the boats is approximately Page 7 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Ex3: A boat leaves San Diego at 30 knots (nautical mph) on a course of 200°. Two hours l ater the boat changes course to 290° for an hour. What is the boat’s bearing and distance from San Diego? Remember: bearing starts N, clockwise Simple Harmonic Motion: describes the motion of objects that oscillate, vibrate, or rotate; can be modeled by the equations d a sin bt or d a cos bt . Frequency = b ; the number of oscillations per unit of time 2 Ex4: A mass on a spring oscillates back and forth and completes one cycle in 3 seconds. Its maximum displacement is 8 cm. Write an equation that models this motion. Period = Amplitude = You Try: You observe a rocket launch from 2 miles away. In 4 seconds, the angle of elevation changes from 3.5° to 41°. How far did the rocket travel and how fast? QOD: What is the difference between an angle of depress Page 8 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 5.1 Using Fundamental Identities + Syllabus Objectives: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (fundamental identities). 3.4 – The student will solve trigonometric equations with and without technology. Identity: a statement that is true for all values for which both sides are defined Example from algebra: 3 x 8 11 3x 13 Simplifying Trigonometric Expressions: Look for identities Change everything to sine and cosine and reduce. Eliminate fractions. Algebra: mulitiply, factor, cancel…. Ex1: Use basic identities to simplify the expressions. a) cot 1 cos2 sin2 cos2 1 sin2 1 cos2 b) tan csc Ex2: a. Simplify the expression (sin x – 1)(sin x + 1) b. Simplify the expression csc x 1 csc x 1 . cos2 x 1 cot 2 csc 2 cot 2 csc 2 1 Use algebra: Ex3: a. Simplify the expression sin x csc x . sin x sin x csc x csc x b. cos (θ – 90°) Simplifying Trigonometric Expressions: Simplify using the following strategies. Note that the equations in bold are the trig identities used when simplifying. All of the other steps are algebra steps. Ex4: Simplify the expression by factoring. sin2 x cos2 x 1 a. b. cos3 x cos x sin 2 x csc2 x cot x 3 Page 9 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities c. sec2 1 d. 4 tan 2 tan 3 sin x cos x Ex5: Simplify the expression by combining fractions. 1 cos x sin x sin2 x cos2 x 1 csc x 1 sin x Verify numerically, graphically. Ex. 6 Rewrite 1 so that it is not in fractional form by Multiplying by the conjugate. 1 sin x Ex 7: Verify the Trigonometric Identity. (Numerically, Graphically) cos 3x 4 cos3 x 3cos x Ex. 8: Use x 2 tan , 0 2 , to write 4 x 2 as a trigonometric function of Reflection: Page 10 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 5.2 Verify Trigonmetric Identities Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities. Trigonometric Identity: an equation involving trigonometric functions that is a true equation for all values of x Tips for Proving Trigonometric Identities: (We are not solving. Do not do anything to both sides.) 1. Manipulate only one side of the equation. Start with the more complicated side. 2. Look for any identities (use all that you have learned so far). 3. Change everything to sine or cosine. 4. Use algebra (common denominators, factoring, etc) to simplify. 5. Each step should have one change only. 6. The final step should have the same expression on both sides of the equation. Note: Your goal when proving a trig identity is to make both sides look identical! For all of the following examples, prove that the identity is true. The trig identities used in the substitutions are in bold. Ex1: cos3 x 1 sin 2 x cos x Ex2: 1 1 2sec2 x 1 sin x 1 sin x Start with the right side (more complicated). sin2 x cos2 x 1 cos2 x 1 sin2 x Start with the left side. Combine fractions. Simplify. Trig substitution. sin2 x cos2 x 1 cos2 x 1 sin2 x Identity 1 sec x cos x Page 11 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Ex3: tan 2 x 1 cos2 x 1 tan 2 x Start with the left side. Trig substitution. tan 2 1 sec 2 Trig substitution. sin2 x cos2 x 1 cos2 x 1 sin 2 x Trig substitution 1 sec x cos x Multiply. Identitiy. sin x tan x cos x Ex4: sec x tan x cos x 1 sin x Start with the left side. Change to sine/cosine. Combine fractions. Multiply num/den by conjugate. Trig substitution. sin2 x cos2 x 1 cos2 x 1 sin2 x Simplify. Ex5: sec 2 1 sin 2 2 sec Start with left side. Split the fraction. Simplify. Trig substitution. sin2 cos2 1 sin2 1 cos2 Identity. 1 cos sec Challenge: Try to prove the identity above in another way. You Try: Prove the identity. cos x sin x 2 cos x sin x 2 2 Reflection: List at least 5 strategies you can use when proving trigonometric identities. Page 12 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 5.3 Solving Trigonmetric Identities Solving Trigonometric Equations Isolate the trigonometric function. Solve for x using inverse trig functions. Note – There may be more than one solution or no solution. Ex1: Solve the equation 4sin 2 x 4 0 in the interval 0,2 . Find values of x for which x sin 1 1 and x sin 1 1 : x Solving Trigonometric Equations: Solve using the following strategies. Find all solutions for each equation in the interval 0,2 . Ex2: Solve the equation by isolating the trig function. 2cos x 1 0 These are values of x where the cosine is equal to 1 . 2 Ex3: Solve the equation by extracting square roots. 4sin 2 x 3 0 These are values of x where the sine is equal to Page 13 of 24 3 . 2 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Ex4: Solve the equation by factoring. 2cos2 x cos x 1 Set equal to zero. Factor. Set each factor equal to zero. Solve each equation. Note: It may be easier to use u-substitution with u cos x to help students visualize the equation as a quadratic equation that can be factored. Ex5: Solve the equation by factoring. 2sec x sin x sec x 0 Factor out GCF. Use zero product property. Solve each equation. Note: It is possible for an equation to have no solution. 2 Ex6: Solve by rewriting in a single trig function. 2sin x 3cos x 3 Substitute Pyth. Identity. sin2 cos2 1 sin2 1 cos2 Simplify algebraically. Factor and solve. Page 14 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Ex7: Solve using trig substitutions. sin 3 x tan x cos x Rewrite sin 3 x sin 2 x sin x Rewrite sin x tan x . cos x Ex. 8 Solve the Function of a multiple angle. 2cos3t 1 0 1. First solve for 3t 2. Then divide the results by 3 Ex9: Find the approximate solution using the calculator. 4 cos x 1 1 Isolate the trig function. cos x 4 1 To find x, we need to find the inverse cosine of ¼. x cos1 4 x ____ When solving an equation in the interval 0,2 , be sure to be in Radian mode. You Try: Make the suggested trigonometric substitution and then use the Pythagorean Identities to write the resulting function as a multiple of a basic trig function. 4 x 2 , x 2cos Reflection: Explain the relationship between trig functions and their cofunctions. Page 15 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 5.4 Sum and Difference Formulas Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (sum and difference identities). Recall: 36 64 100 10 So in general, 36 64 36 64 6 8 14 ab a b and 2 32 52 25 2 32 22 32 13 So in general, (a b)2 a 2 b2 Sum and Difference Identities sin u v sin u cos v cos u sin v cos u v cos u cos v sin u sin v tan u v tan u tan v 1 tan u tan v Note: Be careful with +/− signs! Simplifying Expressions with Sum and Differences 1. Rewrite the expression using a sum/difference identity. 2. Simplify the expression and evaluate if necessary. Ex1: Write the expression as the sine of an angle. Then give the exact value. sin cos cos sin 4 12 4 12 sin u v sin u cos v cos u sin v Evaluating Trigonometric Expressions with Non-Special Angles 1. Rewrite the angle as a sum or difference of two special angles. 2. Rewrite the expression using a sum/difference identity. 3. Evaluate the expression. or Ex2: Find the exact value of cos195 . 195 150 45 cos195 cos u v cos u cos v sin u sin v Page 16 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Ex3: Write as one trig function and find an exact value. tan u v tan80 tan 55 1 tan80 tan 55 tan u tan v 1 tan u tan v Evaluating Trig Functions Given Other Trig Function(s) 15 3 4 , u and sin v , 0 v . 17 2 5 2 We must find cosv and sin u . Ex4: Find cos u v given cos u cos u v cos u cos v sin u sin v Draw the appropriate right triangles in the coordinate plane. 15 3 4 : cos u , u sin v , 0 v : 17 2 5 2 15 u 5 4 17 v Use the Pythagorean Theorem to find the missing sides. In Quadrant III, sine is negative, so sin u _____ . In Quadrant I, cosine is positive, so cos v _____ . 4 15 sin v 17 5 cos u v cos u cos v sin u sin v cos u Page 17 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Proving Identities Ex5: Verify the identity. sin tan tan cos cos Start with the left side. Trig substitution: sin u v sin u cos v cos u sin v Split the fraction: Simplify: Trig substitution: You Try: Verify the cofunction identity sin cos using the angle difference identity. 2 Reflection: Give an example of a function for which f a b f a f b for all real numbers a and b. Then give an example of a function for which f a b f a f b for all real numbers a and b. Page 18 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Date: 5.5 Multiple Angles Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (double angle and power-reducing identities). Ex1: Derive the double angle identities using the sum identities. a.) sin 2u sin u u b.) cos 2u cos u u c.) tan 2u tan u u Double Angle Identities sin 2 2sin cos cos 2 cos 2 sin 2 tan 2 2 tan 1 tan 2 There are two other ways to write the double angle identity for cosine. Use the Pythagorean identity. sin 2 cos 2 1 cos 2 cos2 sin 2 sin 2 1 cos 2 cos 2 1 2sin 2 cos 2 1 sin 2 cos 2 2cos2 1 Page 19 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Evaluating Double-Angle Trigonometric Functions Ex2: Find the exact value of cos 2u given cot u 5, 3 u 2 . 2 5 u 1 u will be in Quadrant IV and forms a right triangle as labeled. Using the Pythagorean Theorem, we have 5 1 cos u , sin u Double Angle Identity: cos 2u cos2 u sin 2 u 26 26 Note: If u is in Quadrant IV, 3 u 2 , then for 2u we have 2 which is in Quadrant IV. So it makes sense that cos 2u is positive. Solving Trigonometric Equations Ex3: Find the solutions to 4sin x cos x 1 in 0,2 . Rewrite the equation. Trig substitution. sin 2 x 2sin x cos x Isolate trig function. Solve for the argument. Because the argument is 2x, we must revisit the domain. 0,2 is the restriction for x. So 0 x 2 . Therefore,. Solve for x. Page 20 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Rewriting a Multiple Angle Trig Function to a Single Angle Ex4: Express sin 3x in terms of sin x . Rewrite argument as a sum Sum identity Double angle identities Pythagorean identity Simplify Verifying a Trig Identity Ex5: Verify sin 2 2 tan . 1 tan 2 Start with left side. Pythagorean identity Rewrite in sines/cosines Simplify Double angle identity Solving for sin 2 and cos2 , we can derive the power reducing identities. cos2 1 2sin 2 sin 2 1 cos2 2 cos2 2cos2 1 cos2 1 cos2 2 1 cos 2 sin 2 1 cos 2 2 tan 2 1 cos 2 1 cos 2 cos 2 2 Page 21 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Power Reducing Identities 1 cos 2 2 1 cos 2 cos 2 2 1 cos 2 tan 2 1 cos 2 sin 2 Ex6: Express cos5 x in terms of trig functions with no power greater than 1. Rewrite as a product Power reducing identity Multiply Power reducing identity You Try: 1. Find the solutions to 2cos x sin 2 x 0 in 0,2 . 2. Verify 2cos2 cot tan . sin 2 Reflection: How do you convert from a cosine function to a sine function? Explain. Page 22 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities 5.5 Half-Angle Identities Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (half angle identities). Recall: sin2 1 cos 2 2 Let u u 1 cos u . We have sin 2 2 2 2 1 cos u u u Solving for sin , we have sin . All of the other half-angle identities can be derived 2 2 2 in a similar manner. Half-Angle Identities sin u 1 cos u 2 2 u 1 cos u cos 2 2 tan u 1 cos u 2 sin u u sin u tan 2 1 cos u tan Note: There are 2 others for tangent. u 1 cos u 2 1 cos u Note: The will be decided based upon which quadrant u lies in. 2 Evaluating Trig Functions Ex1: Find the exact value of a.) cos 12 Rewrite as a half angle Half angle identity 12 is in Quadrant I, where cosine is positive. Evaluate Choose sign b.) tan22.5̊ Page 23 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities Solving a Trig Equation Ex2: Solve the equation sin x sin x in 0,2 . 2 Half-angle identity Square both sides Pythagorean identity Set equal to zero Factor Zero product property You Try: Solve: sin2 x 2 + cos x = 0 Reflection: Explain why two of the half-angle identities do not have +/− signs. Page 24 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4