Date: 4.7 Inverse Trig Functions Syllabus Objective: 4. 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). y sin x Notation: The inverse of 1 f x is labeled as f 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,. Inverse of the Sine Function To graph the inverse of sine, reflect about the line y x. Domain of sin : Range of 1 1 f x x 1 1 f x sin x : Notation: Inverse of Sine or y arcsin x(arcsine) 1 1 f x sin x Note: y 1 sin x denotes the inverse of sine (arcsine). It is NOT the reciprocal of sine (cosecant). Page 1 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Evaluating the Inverse Sine Function 1.. 3. Ex1: Find the exact values of the following. arcsin What value of x makes the equation sin x true? Note: The range of arcsine is restricted to,, so is the only possible answer. 1 sin 3 sin sin 3 1 What value of x makes the equation sin x 3 true? Taking the inverse sine of the sine function results in the argument. 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 cos : Range of 1 1 f x x 1 1 f x cos x : Notation: Inverse of Cosine 1 1 f x cos x or y arccos x(arccosine) 1 Note: y cos x denotes the inverse of cosine (arccosine). It is NOT the reciprocal of cosine (secant). Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Evaluating the Inverse Cosine Function 1.. 3. Ex: Find the exact values of the following. arccos What value of x makes the equation cos x true? Note: The range of arcsine is restricted to 0,, so is the only possible answer. sin cos 1 3 1 11 cos cos 6 cos 1 3, so sin 6 Inverse of the Tangent Function Graph of f x tan x Domain: Range: Domain of tan : Range of 1 1 f x tan x or y arctan x 1 1 f x x Notation: Inverse of Tangent In order for f x tan x to have an inverse function, we must restrict its domain to,. To graph the inverse of tangent, reflect about the line y x. 1 1 f x tan x : (arctangent) 1 Note: y tan x denotes the inverse of tangent (arctangent). It is NOT the reciprocal of tangent (cotangent). Page 3 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Evaluating the Inverse Tangent Function Ex3: Find the exact values of the following. 1. sin tan 1 3 3 1 3 sin tan sin 3 Note: The range of arctangent is restricted to,, so is the only possible answer for 1 3 tan. 3 1 1. costan 1 cos tan 1 cos 3. arccostan arccos tan arccos No Solution, because 3 3 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! 1 Ex4: Evaluate cosarctan without a calculator. 5 Right Triangle θ Hypotenuse: 1 Let arctan. Since the range of arctangent is, 5, and the tangent is positive, must be in Quadrant. Therefore, cosine is positive. So cos. Ex5: Find an algebraic expression equivalent to sin arccos 4x. θ You Try: Evaluate cos 1 cos 4. Be careful! 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 Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
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 1 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 Ex: Two boats lie in a straight line with the base of a cliff 1 meters above the water. The angles of depression are 53 to the nearest boat and 7 to the farthest boat. How far apart are the boats? Distance between the boats is approximately Page 5 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Ex3: A boat leaves San Diego at 30 knots (nautical mph) on a course of 00. Two hours l ater the boat changes course to 90 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 asin bt or d acosbt. b Frequency = ; the number of oscillations per unit of time 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 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 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
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: x 3 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) cot1 cos sin cos 1 sin 1 cos b) tancsc Ex: a. Simplify the expression (sin x 1)(sin x + 1) b. Simplify the expression csc x 1 csc x 1 Use algebra:. cos x 1 cot csc cot csc 1 Ex3: a. Simplify the expression sinxcsc 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. a. cos 3 x cos xsin x sin xcos x 1 b. csc xcot x 3 Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
c. sec 1 d. 4tan tan 3 Ex5: Simplify the expression by combining fractions. Verify numerically, graphically. sin x cos x 1 cos x sin x sin xcos x 1 1 csc x sin x 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) cos3 4cos 3cos x x x Ex. 8: Use x tan, 0, to write 4 x as a trigonometric function of Reflection: Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Date: 5. 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.. 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. Start with the right side (more complicated). sin x cos x 1 cos x 1 sin x 3 Ex1: cos x 1 sin x cos x Start with the left side. Combine fractions. Simplify. Trig substitution. sin x cos x 1 cos x 1 sin x Identity 1 sec x cos x Ex: 1 1 sec 1sin x 1sin x x Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Start with the left side. Trig substitution. tan 1 sec Trig substitution. sin x cos x 1 cos x 1 sin x Trig substitution Multiply. Identitiy. sin x tan x cos x Start with the left side. Change to sine/cosine. Combine fractions. 1 sec x cos x Ex3: tan x 1 cos x 1 tan x cos x Ex4: sec xtan x 1 sin x Multiply num/den by conjugate. Trig substitution. sin x cos x 1 cos x 1 sin x Simplify. Start with left side. Split the fraction. Simplify. Trig substitution. sin cos 1 sin 1 cos Identity. 1 cos sec Ex5: sec 1 sin sec Challenge: Try to prove the identity above in another way. You Try: Prove the identity. x x x x cos sin cos sin Reflection: List at least 5 strategies you can use when proving trigonometric identities. Page 10 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
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 x 4 0 in the interval 0,. 1 1 Find values of x for which x sin 1 and x sin 1 : x Solving Trigonometric Equations: Solve using the following strategies. Find all solutions for each equation in the interval 0,. Ex: Solve the equation by isolating the trig function. cos x 1 0 These are values of x where the cosine is equal to 1. Ex3: Solve the equation by extracting square roots. 4sin x 3 0 These are values of x where the sine is equal to 3. Page 11 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Ex4: Solve the equation by factoring. Set equal to zero. cos xcos x 1 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. sec xsin 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. Ex6: Solve by rewriting in a single trig function. Substitute Pyth. Identity. sin x3cos x 3 sin cos 1 sin 1 cos Simplify algebraically. Factor and solve. Page 1 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Rewrite 3 sin sin sin x x x Ex7: Solve using trig substitutions. 3 sin x tan x cos x Rewrite sin x tan x cos x. Ex. 8 Solve the Function of a multiple angle. cos3t 1 0 1. First solve for 3t. Then divide the results by 3 Ex9: Find the approximate solution using the calculator. 4cos x 1 1 Isolate the trig function. cos x 4 To find x, we need to find the inverse cosine of ¼. x cos 4 1 1 When solving an equation in the interval 0,, be sure to be in Radian mode. x 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, x cos Reflection: Explain the relationship between trig functions and their cofunctions. Page 13 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
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 36 64 36 64 6 8 14 and So in general, a b a b 3 5 5 So in general, ( a b) a b 3 3 13 Sum and Difference Identities sin u v sinu cos v cosu sin v cos u v cosu cos v sinusin v tanu tan v tanuv 1 tanutan v Note: Be careful with +/ signs! Simplifying Expressions with Sum and Differences 1. Rewrite the expression using a sum/difference identity.. 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 1 4 1 sin u v sin ucosv cos usinv Evaluating Trigonometric Expressions with Non-Special Angles 1. Rewrite the angle as a sum or difference of two special angles.. Rewrite the expression using a sum/difference identity. 3. Evaluate the expression. or Ex: Find the exact value of cos195. 195 150 45 cos195 cos u v cos ucosv sin usinv Page 14 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Ex3: Write as one trig function and find an exact value. tan u tanv tanuv 1 tan utanv tan80 tan55 1tan80 tan55 Evaluating Trig Functions Given Other Trig Function(s) Ex4: Find cosu v given 15 3 4 cosu, u and sin v, 0 v. 17 5 cos u v cos ucosv sin usinv We must find cosv and sinu. Draw the appropriate right triangles in the coordinate plane. 15 3 4 cosu, u : sin v, 0 v : 17 5 15 u 17 v 5 4 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. 15 cosu 17 4 sin v 5 cos u v cos ucosv sin usinv Page 15 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Proving Identities Start with the left side. Ex5: Verify the identity. sin tan tan coscos Trig substitution: sin u v sin ucosv cos usinv Split the fraction: Simplify: Trig substitution: You Try: Verify the cofunction identity sin cos using the angle difference identity. 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 16 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
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.) sinu sin u u b.) cosu cos u u c.) tanu tanu u Double Angle Identities sin sin cos cos cos sin tan tan 1 tan There are two other ways to write the double angle identity for cosine. Use the Pythagorean identity. sin cos 1 sin 1 cos cos 1 sin cos cos sin cos 1 sin cos cos 1 Page 17 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Evaluating Double-Angle Trigonometric Functions 3 Ex: Find the exact value of cosu given cot u 5, u. u 5 1 u will be in Quadrant IV and forms a right triangle as labeled. Using the Pythagorean Theorem, we have 5 1 Double Angle Identity: cosu cos u sin u cos u, sin u 6 6 Note: If u is in Quadrant IV, 3 u, then for u we have which is in Quadrant IV. So it makes sense that cosu is positive. Solving Trigonometric Equations Ex3: Find the solutions to 4sin x cos x 1 in 0,. Rewrite the equation. Trig substitution. sinx sin xcos x Isolate trig function. Solve for the argument. Because the argument is x, we must revisit the domain. 0, is the restriction for x. So 0 x. Therefore,. Solve for x. Page 18 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Rewriting a Multiple Angle Trig Function to a Single Angle Rewrite argument as a sum Ex4: Express sin3x in terms of sin x. Sum identity Double angle identities Pythagorean identity Simplify Verifying a Trig Identity Start with left side. tan Ex5: Verify sin. 1 tan Pythagorean identity Rewrite in sines/cosines Simplify Double angle identity Solving for sin and cos, we can derive the power reducing identities. cos 1 sin cos cos 1 1 cos 1 cos sin cos 1 cos sin 1 cos tan cos 1 cos 1 cos Page 19 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Power Reducing Identities 1 cos sin 1 cos cos 1 cos tan 1 cos Ex6: Express Rewrite as a product Power reducing identity Multiply Power reducing identity 5 cos x in terms of trig functions with no power greater than 1. You Try: 1. Find the solutions to cos x sinx 0 in 0,.. Verify cos cot tan sin. Reflection: How do you convert from a cosine function to a sine function? Explain. Page 0 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
5.5 Half-Angle Identities Syllabus Objective: 3.3 The student will simplify trigonometric expressions and prove trigonometric identities (half angle identities). Recall: 1 cos u sin Let. We have u 1 cosu sin u Solving for sin, we have u 1 cosu sin. All of the other half-angle identities can be derived in a similar manner. Half-Angle Identities u 1 cosu sin u 1 cosu cos u 1 cosu tan 1 cos u Note: There are others for tangent. u 1 cosu tan sin u u sinu tan 1 cos u Note: The will be decided based upon which quadrant u lies in. Evaluating Trig Functions Rewrite as a half angle Half angle identity is in Quadrant I, where cosine is positive. 1 Evaluate Choose sign b. tan. Ex1: Find the exact value of a.) cos 1 Page 1 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Solving a Trig Equation Half-angle identity Square both sides Pythagorean identity Set equal to zero Factor Zero product property Ex: Solve the equation sin x in sin x 0,. You Try: x Solve: sin + cos x = 0 Reflection: Explain why two of the half-angle identities do not have +/ signs. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4