Precalculus Notes: Unit 4 Trigonometr Sllabus Objective:. The student will solve problems using the unit circle. Review: a) Convert. hours into hours and minutes. Solution: hour + (0.)(60) = hour and minutes b) Convert hours and 0 minutes into hours. Solution: hours + 0 hours =. hours 60 Bablonian Number Sstem: based on the number 60; 60 approimates the number of das in a ear. Circles were divided into 60 degrees. A degree can further be divided into 60 minutes (60'), and each minute can be divided into 60 seconds (60''). Converting from Degrees to DMS (Degrees Minutes Seconds): multipl b 60 E: Convert 4.59 to DMS. 4.59 = 4 + 0.59(60)' = 4 5.4' = 4 5' +0.4(60)'' = 4 5' 4'' Converting to Degrees (decimal): divide b 60 E: Convert 7 ' 5'' to decimal degrees. 5 7 ' 5'' = 7 7. 60 60 Radian: a measure of length; radian when r s r = radius, s = length of arc θ r s Recall: Circumference of a Circle; C r So, there are radians around the circle radians = 60, or radians = 80 Converting Degrees (D) to Radians (R): E: Convert 540 to radians. 540 radians 80 D R (Note: 80 80 ) Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Note: We multipl b 80 so that the degrees cancel. This will help ou remember what to multipl b. Converting from Radians (R) to Degrees (D): 80 D R d Or use the proportion: 80 r E4: Convert radians to degrees. 4 80 5 4 Note: We multipl b 80 so that the s cancel. Special Angles to Memorize (Teacher Note: Have students fill this in for practice.) Degrees 0 45 60 90 0 5 50 80 Radians Arc Length: s 6 4 r ( must be measured in radians) 5 4 6 E5: A circle has an 8 inch diameter. Find the length of an arc intercepted b a 40 central angle. 4 Step One: Convert to radians. 40 80 Step Two: Find the radius. Step Three: Solve for s. d 8 r 4 4 6 s 4 in Linear Speed (eample: miles per hour): s r l t t length time Angular Speed (eample: rotations per minute): A t angle time Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr E6: Find the linear and angular speed (per second) of a 0. cm second hand. Note: Each revolution generates radians. Linear Speed: A second hand travels half the circumference in 0 seconds: 0. 0. l 0.4.068cm/sec or 0.4.068 cm / sec 0 60 Angular Speed: 80 A radians/sec or 6 / sec 60 0 0 Note: The angular speed does not depend on the length of the second hand! For eample: The riders on a carousel all have the same angular speed et the riders on the outside have a greater linear speed than those on the inside due to a larger radius. E7: Find the speed in mph of 6 in diameter wheels moving at 60 rpm (revolutions per minute). Unit Conversion: 60 rev min 60 min 6 in hr rev ft in mi 580 ft 945 67.47 mph 44 statute (land) mile = 580 feet Earth s radius 956 miles nautical mile = minute of arc length along the Earth s equator E8: How man statute miles are there in a nautical mile? s r r 956 s 956.5miles 80 60 0800 0800 N Bearing: the course of an object given as the angle measured clockwise from due north Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr 5 E9: Use the picture to find the bearing of the ship. 5 N?? 60 5 09 You Tr:. A lawn roller with 0 inch radius wheels makes. revolutions/second. Find the linear and angular speed.. How man nautical miles are in a statute mile? Show our work. QOD: How are radian and degree measures different? How are the similar? Page 4 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objectives:. The student will solve problems using the inverse of trigonometric functions..5 The student will solve application problems involving triangles. Opposite θ Adjacent Hpotenuse Adjacent Opposite Hpotenuse Note: The adjacent and opposite sides are alwas the legs of the right triangle, and depend upon which angle is used. Trigonometric Ratios of θ: opposite adjacent Sine: sin Cosine: cos Tangent: hpotenuse hpotenuse Memor Aid: SOHCAHTOA Reciprocal Functions: opposite tan adjacent Cosecant: hpotenuse csc Secant: opposite hpotenuse sec Cotangent: adjacent adjacent cot opposite Think About It: Which trigonometric ratios in a triangle must alwas be less than? Wh? Sine and cosine, because the lengths of the opposite and adjacent legs are alwas smaller than the length of the hpotenuse. E: For ABC, find the si trig ratios of A. 6 8 4 6 sin A cos A tan A 0 5 0 5 8 4 5 5 4 csc A sec A cot A 4 Teacher Note: It ma help students to label the sides as Opp, Adj, and Hp first. Have students tr finding the si trig ratios for B. B 6 C 0 8 A Special Relationship: sin tan cos Page 5 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr E: Find the other trig ratios given 5 sin. 7 5 opp sin 7 hp Find the missing leg (a): 5 7 α a 5 7 a 4 This is the adjacent leg of angle α. 4 5 7 7 4 cos, tan, csc, sec, cot 7 4 5 4 5 Special Right Triangles: 0-60-90 & 45-45-90 θ (Degrees) 0 45 60 θ (Radians) sinθ cosθ tanθ 6 0 4 45 Teacher Note: Have students fill in the table (un-bolded values). This table must be memorized! Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Evaluating Trigonometric Ratios on the Calculator Note: Check the MODE on our calculator and be sure it is correct for the question asked (radian/degree). Unless an angle measure is shown with the degree smbol, assume the angle is in radians. E: Evaluate the following using a calculator.. sin 4.68 Mode: degree sin 4.68 0.678. sec. Mode: radian Note: There is not a ke for secant. We must use /(cos) because secant is the reciprocal of cosine. sec..760 Also recall: csc and cot sin tan. tan7 5 Mode: degree Convert DMS to decimal degrees on calculator. Note: The degree and minute smbols can be found in the ANGLE menu. The seconds smbol can be found above the + sign (alpha +). tan7 5.0 Inverse Trigonometric Functions: use these to find the angle when given a trig ratio E4: Find θ (in degrees) if 5 cos. cos 5 Mode: degrees 65.76 Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Solving a Triangle: find the missing angles and sides with given information A 5 B E5: Solve the triangle (find all missing sides and angles). C a 5 5 cos A A cos 5.0 mb 90 5.0 6.87 a 4 (Pthagorean Triple: -4-5; or use the Pthagorean Thm) ma5., mb 6.87, a 4 Application Problem E6: A 6 ft in man looks up at a 7 angle to the top of a building. He places his heel to toe 5 times and his shoe is in. How tall is the building? 7 74'' 74'' 5()=689'' Draw a picture: The height of the building is ( + 74) inches. Use trig in the right triangle to solve for : tan7 689 tan7 59.99 689 Height of the building = 74 59.99 74 59.99inches Convert to feet: The building is approimatel 59.99 49.4ft tall. N q m You Tr: Solve the triangle (find all missing sides and angles). M 8 Q QOD: Eplain how to find the inverse cotangent of an angle on the calculator. Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objectives:. The student will solve problems using the unit circle.. The student will solve problems using the inverse of trigonometric functions. Terminal side Initial side Standard Position (of an angle): initial side is on the positive -ais; positive angles rotate counterclockwise; negative angles rotate clockwise Coterminal Angles: angles with the same initial and terminal sides, for eample 60 or E: Find a positive and negative coterminal angle for each.. 0 Positive: 0 60 490 Negative: 0 60 0 7. 7 9 Positive: (Note: An of these are acceptable answers.) Negative: Trigonometric Functions of an Angle II (S) sin (+) tan (+) III (T) r θ cos (+) IV (C) I (A) sin (+) cos (+) tan (+) r r r sin csc r r cos sec r tan cot Memor Aid: To remember which trig functions are positive in which quadrant, remember Awesome Students Take Calculus. A all are positive in QI, S sine (and cosecant) is positive in QII, T tangent (and cotangent) is positive in QIII, C cosine (and secant) is positive in QIV. Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Reference Angle: ever angle has a reference angle, with initial side as the -ais Evaluating Trig Functions Given a Point Use the ordered pair as and Find r: r Check the signs of our answers b the quadrant E: Find the si trig functions of θ in standard position whose terminal side contains the point 5,. Note: The point is in QII, so sine (and cosecant) will be positive. 5,, r 5 4 Evaluating Trig Functions Given an Angle Find the reference angle Label the sides (if the reference angle creates a special right triangle) Check the signs of our answers b the quadrant sin 4 csc 4 cos 5 4 sec 4 5 5 tan cot 5 E: Find the 6 trig functions of 0. To find the reference angle, start at the positive -ais and go counter-clockwise 0. Reference Angle = 0,, r (0-60-90 triangle) sin csc cos sec tan cot QIV, so cosθ is positive. Page 0 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Evaluating Trig Functions Given One Trig Function Determine which quadrant b the signs of the trig functions given Label the sides corresponding to the given trig functions Check the signs of our answers b the quadrant 5 E: Find the si trig functions if sin and tan 0. 7 Since both sine and tangent are positive, we know that θ is in the first quadrant. 5 opp sin r 7 5 7 hp r 4 5 7 sin csc 7 5 cos 4 7 sec 7 4 tan 5 4 cot 4 5 or 5 7 sin csc 7 5 6 7 cos sec 7 6 5 6 tan cot 6 5 Quadrantal Angles: angles with the terminal side on the aes, for eample, 0,90,80,70 Degrees 0 /60 0 45 60 90 80 70 Radians sinθ cosθ tanθ 0/ 6 0 0 4 0 0 0 und 0 und Teacher Note: Have students fill in the table (unbolded cells). Needs to be memorized! E4: Find csc. Find the reference angle: 6 csc csc undefined sin 0 Periodic Functions: A function f t f tc f t for all values of t in the domain of f. is periodic if there is a positive number c such that The period of sine and cosine are and the period of tangent is. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr 60 E5: Evaluate sin without a calculator. 60 600 sin sin sin800 sin The Unit Circle: r Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr You Tr: Find the following without a calculator.. cos990. 5 cot. sin5 QOD: How is the unit circle used to evaluate the trigonometric functions? Eplain. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objectives: 4. The student will sketch the graphs of the si trigonometric functions. 4. The student will graph transformations of the basic trigonometric functions. 4.6 The student will model and solve real-world application problems involving sinusoidal functions. Teacher Note: Have students fill out the table as quickl as the can. Discuss patterns the can use to be able to memorize these. θ 0 0 45 60 90 0 5 50 80 0 5 40 70 60 5 7 5 4 radians 0 6 4 4 6 6 4 sinθ 0 0 0 cosθ 0 0 tanθ 0 und 0 und 0 Sinusoid: a function whose graph is a sine or cosine function; can be written in the form asinbc d Sinusoidal Ais: the horizontal line that passes through the middle of a sinusoid Graph of the Sine Function Use the table above to sketch the graph. Radians: sin Degrees: sin Sine is periodic, so we can etend the graph to the left and right. Characteristics: Domain:, Range:, -intercept: 0,0 -intercepts: 0 k,0 ; k Page 4 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Absolute Ma = Absolute Min = Decreasing:, Increasing:, Graph of the Cosine Function Use the table above to sketch the graph. Period = Sinusoidal Ais: 0 Radians: cos Degrees: cos Cosine is periodic, so we can etend the graph to the left and right. Characteristics: Domain:, Range:, -intercept: 0, -intercepts: k,0; k Absolute Ma = Absolute Min = Decreasing:, Increasing: 0, Period = Sinusoidal Ais: 0 Note: We will work mostl with the graphs in radians, since it is the graph of the function in terms of. Recall: f a h k is a transformation of the graph of f stretch/shrink, h is a horizontal shift and k is a vertical shift.. a represents the vertical Transformations of Sinusoids: General Form asin b h k a: vertical stretch and/or reflection over -ais amplitude = a h: horizontal shift phase shift = h k: vertical shift period = b frequenc = b = number of ccles completed in b Amplitude: the distance from the sinusoidal ais to the maimum value (half the height of a wave) Page 5 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr E: Sketch the graph of 4sin. Vertical stretch: 4 Amplitude = 4 Period: the length of time taken for one full ccle of the wave P b Frequenc: the number of complete ccles the wave completes per unit of time freq = b E: Sketch the graph of sin. Period = Frequenc = Reflection: If a 0, the sinusoid is reflected over the -ais. E: Sketch the graph of cos. Note: Period, amplitude, etc. all sta the same. Vertical Translation: in the sinusoid sin E: Sketch the graph of 4 sin. a b h k, the line k is the sinusoidal ais Sinusoidal Ais: 4 Phase Shift: the horizontal translation, h, of a sinusoid sin a b h k Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr E4: Sketch the graph of cos. Shift cos to the right. Does this graph look familiar? It is sin! Note: Ever cosine function can be written as a sine function using a phase shift. cos sin E5: Sketch the graph of cos4. Then rewrite the function as a sine function. The coefficient of must equal. Factor out an other coefficient to find the actual horizontal shift. cos4 cos 4 4 Phase Shift: Period: 4 4 Graph: Write as a sine function: cos4 sin 4 sin 4 sin 4 8 Writing the Equation of a Sinusoid E6: Write the equation of a sinusoid with amplitude 4 and period that passes through 6,0. Amplitude: a 4 ; Period: b 6 ; b Phase Shift: normall passes through 0,0, so shift right 6 units 4sin 66 or 4sin6 You Tr: Describe the transformations of 7sin 0.5. Then sketch the graph. 4 QOD: How do ou convert from a cosine function to a sine function? Eplain. Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objectives: 4. The student will sketch the graphs of the si trigonometric functions. 4. The student will graph transformations of the basic trigonometric functions. Teacher Note: Have students fill out the table as quickl as the can. θ 0 0 45 60 90 0 5 50 80 0 5 40 70 60 radians 0 sinθ 0 6 4 5 4 6 0 7 5 4 6 4 0 cosθ 0 0 tanθ 0 und 0 und 0 Graph of the Tangent Function Use the table above to sketch the graph. Radians: tan Degrees: tan Characteristics: Domain: k, k Range:, Intercepts: k,0 Alwas Increasing Period: Vertical Asmptotes: k, k sin Note: tan, so the zeros of sine are the zeros of tangent, and the zeros of cosine are the cos vertical asmptotes of tangent. Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Graph of the Cotangent Function Use the table above to sketch the graph. (Cotangent is the reciprocal of tangent.) Radians: cot Degrees: cot Characteristics: Domain: k, k Range:, Intercepts: k,0 Alwas Decreasing Period: Vertical Asmptotes: k, k cos Note: cot, so the zeros of cosine are the zeros of cotangent, and the zeros of sine are the sin vertical asmptotes of cotangent. Graph of the Secant Function Use the table above to sketch the graph. (Secant is the reciprocal of cosine.) Radians: sec Degrees: sec Note: It ma help to graph the cosine function first. Characteristics: Domain: k, k Range:,, Intercept: 0, Local Ma: Local Min: Period: Vertical Asmptotes: k, k Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Graph of the Cosecant Function Use the table above to sketch the graph. (Cosecant is the reciprocal of sine.) Radians: csc Degrees: csc Note: It ma help to graph the sine function first. Characteristics: Domain: k, k Range:,, Intercept: none Local Ma: Local Min: Period: Vertical Asmptotes: k, k Transformations: Sketch the graph of the reciprocal function first! E: Sketch the graph of sec. Graph cos (gra). Shift up, amplitude. Use the zeros of the cosine function to determine the vertical asmptotes of the secant function. Sketch the secant function (black). Note: Amplitude is not applicable to secant. The number represents a vertical stretch. E: Sketch the graph of tan. Period: P Reflect over -ais Page 0 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Solving a Trigonometric Equation Algebraicall E: Solve for in the given interval using reference triangles in the proper quadrant. sec, Quadrant II; cos Reference Angle = In QII: Solving a Trigonometric Equation Graphicall E4: Use a calculator to solve for in the given interval: cot 5, 0 The equation cot 5 is equivalent to the equation tan. Graph each side of the equation and find 5 the point of intersection for 0. Restrict the window to onl include these values of. Note: We could have also graphed and 5. (Shown in the second graph above.) tan Solution: 0.97 You Tr: Graph the function csc. QOD:. What trigonometric function is the slope of the terminal side of an angle in standard position? Eplain our answer using the unit circle.. Which trigonometric function(s) are odd? Which are even? Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objective: 4.4 The student will sketch the graph of combined trigonometric functions (sum, difference, product or composition of two trigonometric or polnomial functions) with or without technolog. Trigonometric Functions Combined with Algebraic Functions Recall: Periodic Functions: A function f t is periodic if there is a positive number c such that f tc f t for all values of t in the domain of f. Graph the following and determine which function(s) are periodic.. cos Not Periodic. cos Not Periodic. cos Periodic 4. cos Not Periodic Absolute Value of Trig Functions E: Sketch the graph of f cos. The absolute value makes all of the -values of the function positive. Note: The period of f cos is. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Adding a Sinusoid to a Linear Function E: Analze the graph of sin. Is it periodic? The function oscillates between the lines. It is NOT periodic. and Are the Sums of Sinusoid Function Sinusoids? Teacher Note: Have students graph various sums to determine a pattern. Conclusion: The sum of two sinusoids is a sinusoid if and onl if the two sinusoids being added have the same period. The sum has this same period as well. E: Graph the function f sin cos. Write a sinusoid function that estimates f. The period of f is the same as each of the addends. There appears to be a phase shift and an amplitude. Use the zero and ma features on the calculator. Phase Shift: left 0.6 Amplitude: a =.6 Sinusoid: f.6sin 0.6 Note: Check our answer b graphing. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Damped Oscillation: the amplitude is reduced in the wave of a graph (the amplitude decreases as time increases The graph of the product of a function, f, and a sinusoid oscillates between the graph of the function, f and its opposite, f. The function f is called the damping factor. E4: Graph the function f. sin4 damping occur?. Identif the damping factor. Where does the Damping Factor:. Occurs as You Tr: Does the function f sin cos represent a sinusoid? If es, write an equation for the sinusoid. QOD: Are all sinusoids periodic? Are all periodic functions sinusoids? Eplain. Page 4 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objective: 4. The student will sketch the graphs of the principal inverses of the si 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. f is labeled as Inverse of the Sine Function Graph of f sin In order for f sin Domain:, Range:, to have an inverse function, we must restrict its domain to,. Domain of Notation: Inverse of Sine To graph the inverse of sine, reflect about the line sin :, Range of f f sin f sin or arcsin (arcsine). :, Note: sin denotes the inverse of sine (arcsine). It is NOT the reciprocal of sine (cosecant). Evaluating the Inverse Sine Function... E: Find the eact values of the following. arcsin What value of makes the equation sin true? Note: The range of arcsine is restricted to,, so is the onl possible answer. 4 sin sin sin What value of makes the equation sin true? No solution, because > and the domain of arcsine = the range of sine = Taking the inverse sine of the sine function results in the argument. 4, Page 5 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Inverse of the Cosine Function Domain:, Range:, to have an inverse function, we must restrict its domain to Graph of f cos In order for f cos 0,. Domain of Notation: Inverse of Cosine To graph the inverse of cosine, reflect about the line. f cos, f cos 0, : Range of : f cos or arccos (arccosine) Note: (secant). cos denotes the inverse of cosine (arccosine). It is NOT the reciprocal of cosine Evaluating the Inverse Cosine Function... E: Find the eact values of the following. arccos What value of makes the equation cos true? 4 Note: The range of arcsine is restricted to 0,, so is the onl possible answer. 4 sin cos cos, so 6 sin 6 cos cos 6 Taking the inverse cosine of the cosine function results in the argument. 6 Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Inverse of the Tangent Function Graph of f tan In order for f tan Domain: k Range:, to have an inverse function, we must restrict its domain to,. Domain of To graph the inverse of tangent, reflect about the line tan :, f Notation: Inverse of Tangent Range of f tan f tan. :, or arctan (arctangent) Note: tan denotes the inverse of tangent (arctangent). It is NOT the reciprocal of tangent (cotangent). Evaluating the Inverse Tangent Function. E: Find the eact values of the following. sintan sintan sin 6 Note: The range of arctangent is restricted to,, so is the onl possible answer for tan. 6. costan. arccostan cos tan cos 4 arccos tan arccos Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4 No Solution, because. Right Triangle Trigonometr 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 (Pthagorean Theorem) Evaluate the trig function be sure to choose the correct sign!
Precalculus Notes: Unit 4 Trigonometr E4: Evaluate cosarctan 5 without a calculator. 6 θ Right Triangle 5 Hpotenuse: 5 6 Let arctan. Since the range of arctangent is, 5, and the tangent is positive, must be in 5 Quadrant I. Therefore, cosine is positive. So cos. 6 b E5: Find an algebraic epression equivalent to sin arccos4 θ 4. 4 6 b 6 sin arccos 4 6 You Tr: Evaluate cos cos 4. Be careful! QOD: Eplain how the domains of sine, cosine, and tangent must be restricted in order to create an inverse function for each. Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr Sllabus Objective: 4.5 The student will model real-world application problems involving graphs of trigonometric functions. Angle of Elevation: the angle through which the ee moves up from horizontal to look at something above Angle of Depression: the angle through which the ee moves down from horizontal to look at something below Angle of Elevation Angle of Depression Solving Application Problems with Trigonometr: 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 our calculator is in the correct Mode (degrees/radians). E: If ou stand feet from a statue, the angle of elevation to the top is 0, and the angle of depression to the bottom is 5. How tall is the statue? 0 5 tan0 6.98 Height of the statue is approimatel 6.98.5 0.4 ft tan5 tan5.5 E: Two boats lie in a straight line with the base of a cliff meters above the water. The angles of depression are 5 to the nearest boat and 7 to the farthest boat. How far apart are the boats? 5 7 m? = Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr tan5 5.85 tan5 tan 7 4.5 tan 7 Distance between the boats is approimatel 4.5 5.85 5.9m E: A boat leaves San Diego at 0 knots (nautical mph) on a course of 00. Two hours later the boat changes course to 90 for an hour. What is the boat s bearing and distance from San Diego? 0 70 0 mi θ 0 0 60 mi 0 70 9 80 90 Remember: bearing starts N, clockwise At 0 knots for hours, the boat travels 60 nautical miles. At 0 knots for hour, the boat travels 0 nautical miles. Bearing: Distance: 0 60 tan tan 6.565 d 0 60 67.08 nautical miles Bearing = 00 6.565 6.565 Simple Harmonic Motion: describes the motion of objects that oscillate, vibrate, or rotate; can be modeled b the equations d asinbt or d acosbt. b Frequenc = ; the number of oscillations per unit of time E4: A mass on a spring oscillates back and forth and completes one ccle in seconds. Its maimum displacement is 8 cm. Write an equation that models this motion. Period = : b Amplitude = 8 b d 8sin t Page 0 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4 Trigonometr You Tr: You observe a rocket launch from miles awa. In 4 seconds, the angle of elevation changes from.5 to 4. How far did the rocket travel and how fast? QOD: What is the difference between an angle of depression and an angle of elevation? Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter 4