Pre-Calculus. *UNIT 3* Chapter 4 Trigonometric Functions. Name

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1 Pre-Calculus *UNIT 3* Chapter 4 Trigonometric Functions Name FALL

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3 Section 4.1 Notes: Right Triangle Trigonometry Objectives: - find values of trig functions in right triangles and solve right triangles. Remember SOHCAHTOA???? Eample 1: Find the eact values of the si trigonometric functions of θ. Eample 2: If sinθ = 1, find the eact value of the five remaining trigonometric functions for the acute angle θ. 3 Using trigonometry to find a missing side Eample 3: Find the value of. Round to the nearest tenth, if necessary. 2

4 Eample 4: A competitor in a hiking competition must climb up the inclined course as shown to reach the finish line. Determine the distance in feet that the competitor must hike to reach the finish line. Using trigonometry to find a missing angle sin θ = Eample 5: Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary. Angles of Elevation and Depression Angle of elevation: angle formed by a horizontal line and an observer s line of sight to an object ABOVE. Angle of depression: angle formed by a horizontal line and an observers line of sight to an object BELOW. Eample 6: a) The chair lift at a ski resort rises at an angle of while traveling up the side of a mountain and attains a vertical height of 1200 feet when it reaches the top. How far does the chair lift travel up the side of the mountain? 3

5 b) A person on an airplane looks down at a point on the ground at an angle of depression of 15. The plane is flying at an altitude of 10,000 feet. How far is the person from the point on the ground to the nearest foot? Eample 7: A sightseer on vacation looks down into a deep canyon using binoculars. The angles of depression to the far bank and near bank of the river below are 61 and 63, respectively. If the canyon is 1250 feet deep, how wide is the river? Solving Right Triangles - Find the measures of all the and of the triangle Eample 8: Solve ΔFGH. Round side lengths to the nearest tenth and angle measures to the nearest degree. Common Angles 4

6 Time to Hand Jive the TRIGONOMETRY HAND JIVE 1. Use your LEFT Hand, palm open towards you 2. Since all original trig functions are divided by 2, put an invisible 2 in Label each finger according to the following degrees: your palm. 3. Bend in the finger of the specified angle. 4. COSINE: fingers ABOVE bent finger; SINE: fingers BELOW bent For eample 30, bend in your ring finger. finger. Take the square root of the amount of fingers. 1 finger below 5. Evaluate any sin or cos function of common angles. 6. TANGENT: Flip your hand over. Since tan = fpngpps fppm sin numppatpp fpngpps fppm cos numppatpp. Rationalize. sin cos, divide Eample 9: Find eact value without a calculator. a) sin 45º b) cos 30º c) tan 60º d) sec 60º e) cot 30º f) csc 45º Eample 10: Find θ without a calculator. a) sin θ = 1 2 b) csc θ = c) cot θ = 3 d) sec θ = 2 5

7 4-1 Practice Right Triangle Trigonometry Find the eact values of the si trigonometric functions of θ Find the value of. Round to the nearest tenth, if necessary On a college campus, the library is 80 yards due east of the dormitory and the recreation center is due north of the library. The college is constructing a sidewalk from the dormitory to the recreation center. The sidewalk will be at a 56 angle with the current sidewalk between the dormitory and the library. To the nearest yard, how long will the new sidewalk be? 6. If cot A = 8, find the eact values of the remaining trigonometric functions for the acute angle A. Find the measure of angle θ. Round to the nearest degree, if necessary Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree SWIMMING The swimming pool at Perris Hill Plunge is 50 feet long and 25 feet wide. If the bottom of the pool is slanted so that the water depth is 3 feet at the shallow end and 15 feet at the deep end, what is the angle of elevation at the bottom of the pool? 6

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9 Objectives: - Convert degrees to radians and vice versa - Use angle measures to solve real world problems Section 4.2 Notes Degrees and Radians Angles and their Measure Angle: two that share a verte Initial side: point of the angle Terminal side: of the angle Standard Position: verte at, initial side on. Degrees The most common unit for measuring angles Full rotation: 360 Common measures (in diagram) Radian Measure 8

10 Conversions Eample 2: a) Write 135 in radians as a multiple of π (in terms of π) b) Write 2π 3 in degrees Coterminal Angles - Angles that have the same terminal side, but different measures Degrees: α ± 360n Radians: α ± 2πn Eample 3: Identify all angles that are coterminal with the given angle. Then find specifically one positive and one negative coterminal angle. a) 80 b) π 4 Arc Length Eample 4: Find the length of the intercepted arc in a circle with a central angle measure and radius given. Round to nearest tenth if necessary. a) Central angle: π, radius: 4 in b) Central angle: 125, radius: 7 cm 3 9

11 Speed Linear Speed: the rate at which an object moves along a circular path (measured in units simlar to miles per hour) Angular Speed: the rate at which the object rotates about a fied point (measured in units similar to revolutions per minute) As long as you have the same units, you can use v = ϖ r Eample 5: A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at minute. revolutions per a) Find the angular speed, in radians per minute, of a record as it plays. Round to the nearest tenth. b) Find the linear speed at the outer edge of the record as it spins, in centimeters per second. Area of a Sector Sector: a region bounded by a central angle and its intercepted arc. Eample 6: Find the area of the shaded sector of the circle. a) b) 10

12 4.2 Practice Degrees and Radians Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth ' 10" ' 35" Write each degree measure in radians as a multiple of π and each radian measure in degrees π π 3 Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle π 4 Find the length of the intercepted arc with the given central angle measure in a circle of the given radius. Round to the nearest tenth , r = 8 yd 12. 7π, r = 10 in. 6 Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation. 13. ω = 4 π rad/s 14. V = 32 m/s, 100 rev/min On a game show, a contestant spins a wheel. The angular speed of the wheel was ω = π radians per second. If the wheel 3 maintained this rate, what would be the rotation in revolutions per minute? Find the area of each sector. 16. θ = π 7π, r = 14 in. 17. θ =, r = 4 m

13 Section 4.3 Notes Trigonometry Functions on the Unit Circle Objectives: - Find the values of trig functions for any angle more than just Positive Acute Angles Eample 1: Let ( 4, 3) be a point on the terminal side of an angle, θ, in standard position. Find the eact value of the si trigonometric functions of θ. Quadrantal Angles - When the terminal side of and angle θ that is in standard position lies on one of the coordinate aes. Eample 2: Find the eact value. If not define, write undefined. a) cos π b) tan 450 c) cot 7π 2 12

14 Reference Angles - Denoted θ - Positive acute angle measure between a terminal side and the - ais. Eample 3: Sketch the angle. Then find its reference angle. a) -150 b) 3π 4 c) 520 Evaluating using Reference Angles -the trig values of an angle and its reference angle are the SAME, with the eception of being positive or negative Positive/Negative Rules Eample 4: Find the eact value. a) sin 4π 3 b) tan 150 c) sec 15π 4 13

15 Eample 5: Let csc θ = 3, tan θ < 0. Find the eact values o the five remaining trigonometric functions of θ. Eample 6: A student programmed a 10-inch long robotic arm to pick up an object at point C and rotate through an angle of 150 in order to release it into a container at point D. Find the position of the object at point D, relative to the pivot point O. Unit Circle -a circle of radius 1, centered at the origin -an angle θ in radians is equivalent to arc length, s Eample 7: Find the eact value. If not define, write undefined. a) sin 7π 6 b) tan 4π 3 c) sec 270 Eample 8: Find the eact value of all si trigonometric functions for θ = 5π 6. 14

16 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Periodic Functions -functions that repeat at regular intervals *Rewrite a given degree/radian as the sum of a number and a full revolution (360 or 2π) Eample 9: Find the eact value. a) cos 9π 4 b) sin (-300 ) c) tan 29π 6 15

17 4-3 Practice Trigonometric Functions on the Unit Circle The given point lies on the terminal side of an angle θ in standard position. Find the values of the si trigonometric functions of θ. 1. ( 1, 5) 2. (7, 0) 3. ( 3, 4) 4. (1, 2) 5. ( 3, 1) 6. (2, 4) Sketch each angle. Then find its reference angle π π π π 3 Find the eact value of each epression. If undefined, write undefined. 13. csc tan sin ( 90 ) 16. cos 3π Sec π Cot 5π PENDULUMS The angle made by the swing of a pendulum and its vertical resting position can be modeled by θ = 3 cos πt, where t is time measured in seconds and θ is measured in radians. What is the angle made by the pendulum after 6 seconds? 16

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19 Objective - Graph transformations of the sine and cosine function Section 4.4 Notes Graphing Trigonometric Functions Basics of Sine and Cosine Notice the cosine graph is a horizontal translation of the sine graph. Any transformation of a sine function is called a SINUSOID A portion of the curve represents one period, or one cycle. (One hill and one valley) Amplitude - Denoted a - y = asin or y = acos - Vertical Epansion when - Vertical Compression when For y = a sin b( h) + k y = a cos b( h) + k, amplitude = a Eample 1: Describe how the graphs of f () = sin and g () are related. Then find the amplitude of g (). a) g() = 2.5 sin b) g () = 0.4 cos 18

20 Reflections - When a is - Reflection in the Eample 2: Describe how the graphs of f () = cos and g () = 2 cos are related. Then find the amplitude of g (). Period - Horizontal Epansion when - Stretch out a cycle - Horizontal Compression when - Squeeze a cycle together For y = a sin b( h) + k y = a cos b( h) + k, period = 2π b Eample 3: Describe how the graphs of f () = cos and g () are related. Then find the period of g (). a) g() = cos 8 b) g () = sin 4 Frequency For y = a sin b( h) + k y = a cos b( h) + k, frequency = 1 = b pppppp 2π 19

21 Horizontal/Vertical Shift y = sin ( h) or y = cos ( h) h is a horizontal (always in parenthesis and always opposite!) y = sin + k or y = cos + k k is a **when h is present, b MUST be outside of the parenthesis. If it is not, to make the coefficient of = 1. EX(i): y = sin(2 +6) + 3 EX (ii): y = cos ( ) 9 Eample 5: State the amplitude, period, frequency, and phase shift translations of a) y = 2 sin(4 + π 4 ) b) y = 4 cos( 3 + π 6 ) Eample 6: State the amplitude, period, frequency, and phase shift translations of a) y = sin( + π) + 1 b) y = 3sin(4 + π 2 ) + 1 SUMMARY The graphs y = a sin b( h) + k and y = a cos b( h) + k, where a 0 and b 0, have the following characteristics: Horizontal shift: h k PRACTICE State the amplitude, period, frequency, phase shift translation, and vertical shift of each function. 1. y = 2 sin + π 3 2. y = 1 cos (2 π)

22 GRAPHING y = a sin b( h) + k and y = a cos b( h) + k a multiply the by this number. b multiply the by the reciprocal of this number. h add this number to the k add this number to the Perform a and b First Perform h and k Net The 5 Important Points Sine: Cosine: (0, 0) (0, 1) π, 1 2 π, 0 2 (π, 0) (π, -1) 3π, 1 2 3π, 0 2 (2π, 0) (2π, 1) To find the increments of the -ais Find the period of the function. Evaluate: pppppp 4 To label the first positive tick mark, add this value to 0. Continue this pattern until all marks are labeled. To label the second pos. tick mark, add the value to the first tick mark. Midline: the sine and cosine curves oscillate about a central line that goes through the curve. With no shifts, the midline is the ais. With shifts, the midline is the horizontal line. Eample 7: Sketch 2 periods of the function. a) y = 3 sin 2 π 2 a = b = h = k = Original a / b shift h / k shift b) y = 3 cos 2π + 2 a = b = h = k = Original a / b shift h / k shift 21

23 c) y = 2 sin π ( + 1) 1 2 a = b = h = k = Original a / b shift h / k shift d) y = 2 sin(4 + π 2 ) a = b = h = k = Original a / b shift h / k shift e) y = 2 cos + π 2 +1 a = b = h = k = Original a / b shift h / k shift 22

24 f) y = 1 cos (2 π) a = b = h = k = Original a / b shift h / k shift MATCHING Match the equation to the correct lettered picture. 23

25 Section 4.4 Day 2 GRAPHING HOMEWORK 14. a = b = h = k = Original a / b shift h / k shift 15. a = b = h = k = Original a / b shift h / k shift 16. a = b = h = k = Original a / b shift h / k shift 24

26 17. a = b = h = k = Original a / b shift h / k shift 18. a = b = h = k = Original a / b shift h / k shift 19. a = b = h = k = Original a / b shift h / k shift 25

27 4-4 Practice Graphing Sine and Cosine Functions Describe how the graphs of f() and g() are related. Then find the amplitude of g() and sketch two periods of both functions on the same coordinate aes. 1. f() = sin 2. f() = cos g() = 1 3 sin g() = 1 4 cos State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of the function. 3. y = 2 sin + π 3 4. y = 1 cos (2 π) Write a sinusoidal function with the given amplitude, period, phase shift, and vertical shift. 5. sine function: amplitude = 15, period = 4π, phase shift = π, vertical shift = cosine function: amplitude = 2, period = π, phase shift = π, vertical shift =

28 7. Sketch two periods of the function. π y = 4 1 a) y = 2 sin 2 b) 3 cos ( 2) c) y = 2 sin + 1 d) y = cos( 2 + π )

29 Section 4.5 Notes Graphing other Trigonometric Functions Objective: - Graph tangent and reciprocal functions Tangent Function Remember, y = tan is equivalent to Determine vertical asymptotes by solving: and y = a tan (b + c) + d a produces a vertical stretch or (a 0) - Negative a values reflect in the b affects the period (b 0) c produces a d produces a Eample 1: a) Locate the vertical asymptotes and sketch the graph of y = tan 3. y 28

30 b) Locate the vertical asymptotes of the graph of y = 4tan y c) Sketch the graph of y = 4tan. Eample 2: Locate the vertical asymptotes and sketch the graph y a) y = -tan 4. b) y = -tan( + π 2 ). y c) Locate the vertical asymptotes of the graph of y = -tan(3 + π). 29

31 Cotangent Function Remember, y = cot is equivalent to Determine vertical asymptotes by solving: and Eample 3: a) Locate the vertical asymptotes and sketch the graph of y = cot 2. b) Locate the vertical asymptotes of the graph of y = 1 cot 2. 2 Secant and Cosecant Functions Remember, y = csc is equivalent to and y = sec is equivalent to 30

32 Vertical Asymptotes Cosecant: Solve and Secant: Solve and Eample 4: Locate the vertical asymptotes and sketch the graph. y a) y = -sec 2 b) y = csc π. 2 y 31

33 4-5 Practice Graphing Other Trigonometric Functions Locate the vertical asymptotes, and sketch the graph of each function. 1. y = 3 tan 2. y = 2 cot 2 + π 3 3. y = csc y = sec 3 + π 1 32

34 Section 4.5 GRAPHING HOMEWORK (Use for day 1 and day 2) y y y y y y 33

35 Objectives: - Evaluate and Graph inverse trig functions - Find compositions of trig functions Section 4.6 Notes: Inverse Functions Inverse SINE Is the sine function one to one? The domain must be restricted to the interval to make the function one to one. The inverse equation is y = sin -1 and is graphed by reflecting the restricted y = sin in the line y =. Notice the domain of the inverse is and its range is Because angles and arcs given on the unit circle have equivalent radian measures, the inverse sine function is sometimes referred to as the arcsine function y = arcsin. sin -1 or y = arcsin can be interpreted as the angle (or arc) between π and π with a sine of. 2 2 For eample, sin is the angle with a sine of 0.5. Recall that sin t is the y-coordinate of the point on the unit circle that corresponds to the angle or arc length, t. Because the range of the inverse sine function are restricted, the possible angle measures of the inverse sine function are located on the half of the unit circle Eample 1: Find the eact value, if it eists. 1 2 a) sin 2 b) arcsin 3 2 c) sin 1 2π d) sin

36 Inverse COSINE To make the cosine function one-to-one, the domain must be restricted to. The inverse cosine function is y = cos -1 or arccosine function The graph of y = cos -1 is found by reflecting the graph of the restricted y = cos in the line y =. Recall that cos t is the -coordinate of the point on the unit circle that corresponds to the angle or arc length, t. Because the range of y = cos -1 is restricted to, the possible angle measures of the inverse cosine function are located on the of the unit circle Eample 2: Find the eact value, if it eists. a) cos 1 1 b) arccos 3 2 c) cos 1 2 d) cos 1 1 Inverse TANGENT To make the tangent function one-to-one, the domain must be restricted to ( π 2, π 2 ). The inverse tangent function is y = tan -1 or arctangent function y = arctan. The graph of y = tan -1 is found by reflecting the graph of the restricted y = tan in the line y =. *Unlike sine and cosine, the domain of the inverse tangent function is sin On the unit circle, tant = or tant = y cos The values of y = tan -1 will be located on the of the unit circle, not including π and π because the 2 2 tangent function is undefined at those points. 35

37 Eample 3: Find the eact value, if it eists. 1 3 a) tan 3 b) arctan 1 c) arctan ( 3) INVERSE SUMMARY Graphing Inverse Trigonometric Functions Rewrite the function in one of the following forms: Make a table of values, assigning radians from the restricted range to y. Plot the points and connect them with a smooth curve. Eample 4: Sketch the graph. a) y = arctan 2 y b) y = sin 1 2 y 36

38 y c) y = arccos 4 Eample 5: a) In a movie theater, a 32-foot-tall screen is located 8 feet above ground. Write a function modeling the viewing angle θ for a person in the theater whose eye-level when sitting is 6 feet above ground. b) In a movie theater, a 32-foot-tall screen is located 8 feet above ground-level. Determine the distance that corresponds to the maimum viewing angle. Composition of Trigonometric Functions In Lesson 1.7, you learned that if is in the domain of f() and f -1 () then f [f -1 ()] = and f -1 [f()] = Because the domains of the trig functions are restricted to obtain the inverse trig function, the properties do not apply for all values of. For eample, while sin is defined for all, the domain of sin -1 is [-1,1]. Therefore, sin(sin -1 ) = is only true when A different restriction applies for the composition of sin -1 (sin) because the domain of sin is [ π 2, π 2 ]. Therefore, sin -1 (sin) = is only true when 37

39 Summary of Domain Restrictions Eample 6: Find the eact value, if it eists. a) sin arcsin 1 2 b) cos 1 cos 5π 2 c) arctan tan 5π 2 d) arcsin sin 7π 6 Eample 7: Find the eact value. a) sin cos b) cos arcsin c) tan arccos Reciprocal Inverses arccosecant arcsecant arccotangent Etra Eample: Find the eact value. a) csc cot b) tan csc 1 c) sec 4 5 cos 1 15 d) cot 17 sec

40 Eample 8: a) Write cot (arccos ) as an algebraic epression of that does not involve trigonometric functions. b) Write cos(arctan ) as an algebraic epression of that does not involve trigonometric functions. PART OF THE HOMEWORK: sin tan csc cos sin tan 4 39

41 4-6 Practice Inverse Trigonometric Functions Sketch the graph of each function. 1. y = arccos 3 2. y = arctan y = sin y = tan 1 3 Find the eact value of each epression, if it eists. 5. arcsin cos 1 cos π 3 7. tan 3π 2 8. sin 1 cos π 3 9. arctan arcsin tan sin 1 1 cos sin arctan ART Hans purchased a painting that is 30 inches tall that will hang 8 inches above the fireplace. The top of the fireplace is 55 inches from the floor. a. Write a function modeling the maimum viewing angle θ for the distance d for Hans if his eye-level when sitting is 2.5 feet above the ground. b. Determine the distance that corresponds to the maimum viewing angle. 40

42 Write each trigonometric epression as an algebraic epression of. 14. sin (arccos ) 15. tan (sin 1 ) Find the eact value of the epression without a calculator, if it eists tan Cos sec (sin 1 ( 0.5)) cos sin tan cos tan cos 4 41

43 Section 4.7 Law of Sines and Law of Cosines, Area, and Application Objectives: - Solve oblique triangles using Law of Sines and Law of Cosines - Find areas of oblique triangles. Oblique Triangle a triangle that is not a right triangle Law of Sines Used to solve a triangle when the 3 measurements provided fit one of these cases: * Two angles and a nonincluded side * Two angles and the included side * Two sides and the included angle *choose any 2 ratios to create a proportion with 3 known measurements, and 1 unknown. Eample 1: Solve the given triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. a) b) Eample 2: a) The angle of elevation from the top of a building to a hot air balloon is 62º. The angle of elevation to the hot air balloon from the top of a second building that is 650 feet due east is 49º. Find the distance from the hot air balloon to each building. 42

44 b) A tree is leaning 10 past vertical as shown in the figure. A wire that makes a 42 angle with the ground 10 feet from the base of the tree is attached to the top of the tree. How tall is the tree? Solving SSA cases * It is possible for more than 1 triangle to eist, or NO triangle to eist. It all depends on if the given angle is. * Always look for 2 triangles when finding an angle using Law of Sines. The Ambiguous Case Given an Obtuse *if OPP ADJ, then NO eists *if OPP > ADJ, then ONE eists Given an Acute *if ADJsin( ) > OPP, then NO eists *if ADJsin( ) = OPP, then ONE eists *if OPP > ADJ, then ONE eists *if ADJsin( ) < OPP < ADJ, then TWO s eist How to Find the 2 nd s Answers 1. Using the FIRST angle you found, subtract this number from 180º. This is the 2 nd degree measure for the same angle. 2. The ORIGINAL angle given in the problem, will never change, so take this angle AND the angle from step 1 and subtract them from 180º. (Since all 3 angles have to add up to 180º.) 3. Use Law of Sines with your new angles to find the remaining side. 43

45 Eample 3: Find all solutions for the given triangle, if possible. If no solution eists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. a) m A = 63, a = 18, b = 25 b) m C = 105, b = 55, c = 73 Eample 4: Find all solutions for the given triangle, if possible. If no solution eists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. a) m B = 45, b = 18, and c = 24 b) m C = 24, c = 13, and a = 15 Law of Cosines When the 3 measurements provided fit one of these cases: * Three sides (SSS) * Two sides and the included angle (SAS) Eample 5: a) A triangular area of lawn has a sprinkler located at each verte. If the sides of the lawn are a = 19 feet, b = 24.3 feet, and c = 21.8 feet, what angle of sweep should each sprinkler be set to cover? 44

46 b) A triangular lot has sides of 120 feet, 186 feet, and 147 feet. Find the angle across from the shortest side. c) Two airplanes leave an airport at the same time on different runways. One flies on a bearing of N66 W at 325 miles per hour. The other airplane flies on a bearing of S26 W at 300 miles per hour. How far apart will the airplanes be after two hours? d) Two airplanes leave an airport at the same time on different runways. One flies directly north at 400 miles per hour. The other airplane flies on a bearing of N75 E at 350 miles per hour. How far apart will the airplanes be after two hours? Solving a using Law of Cosines *In SSS, you must find the angle first, then use the Law of Sines to find the of the 2 remaining angles. *In SAS, you must first find the from the, then use the Law of Sines to find the of the 2 remaining angles. Eample 6: Solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. a) b) Solve ΔMNP if m M = 54 o, n = 17, and p = 12. c) 45

47 Area of Oblique Triangles Eample 8: Find the area of the triangle to the nearest tenth. a) b) Find the area of the quadrilateral 1) Divide the quadrilateral into, in a way that creates a triangle with SAS information. Find the area of this triangle using A = ½ absinc. 2) Name the side the triangles share. Use to solve for. 3) Use the given angle and to find another angle in this triangle using. (Find the partial angle of the angle that got split) 4) Find the remaining degree in the 2 nd triangle and then find the area of the 2 nd triangle using. 5) Add the areas together to find the total area

48 Eample 9: Find the area of the plot of land. a) A post is driven in a certain spot. Proceed due east for 300 ft, then proceed S40 o E for another 150 ft. Turn direction again S60 o W for 400 ft and then back to the post in a straight line. Sketch the figure and find its area. P b) From an iron post, proceed 500 m northeast to the brook, then 300 m east along the brook to the old mill, then 200 m S15 E to a post on the edge of Wiggin s Road, and finally along Wiggin s Road back to the iron post. Sketch the figure and find its area. P 47

49 4-7 Practice The Law of Sines and the Law of Cosines Solve each triangle. Round to the nearest tenth if necessary STREET LIGHTING A lamp post tilts toward the Sun at a 2 angle from the vertical and casts a 25-foot shadow. The angle from the tip of the shadow to the top of the lamp post is 45. Find the length of the lamp post. Find the area of each triangle to the nearest tenth. 10. RST if R = 115, s = 15 yd, t = 20 yd 11. MNP if n = 4 ft, P = 69, N = DEF if d = 2 ft, E = 85, F = JKL if j = 68 cm, l = 110 cm, K =

50 Pre-Calculus More Survey Problems Name: 1. Find the area of the quadrilateral. Throughout the problem, round side lengths to the nearest tenth, angle measures to the nearest degree, and the final area to the nearest square foot. 112 ft 98 ft ft Total Area = 2. Find the area of the quadrilateral. Throughout the problem, round side lengths to the nearest tenth, angle measures to the nearest degree, and the final area to the nearest square inch. 13 in. 11 in in. Total Area = 49

51 3. A post is driven in a certain spot. Proceed due east for 600 ft, then proceed S20 o E for another 250 ft. Turn direction again S40 o W for 500 ft and then back to the post in a straight line. Sketch the figure and find its area. Throughout the problem, round side lengths to the nearest tenth, angle measures to the nearest degree, and the final area to the nearest square foot. P w y z Total Area = 4. From an iron post, proceed 1000 m northeast to the brook, then 500 m east along the brook to the old mill, then 400 m S22 E to a post on the edge of Wiggin s Road, and finally along Wiggin s Road back to the iron post. Sketch the figure and find its area. Throughout the problem, round side lengths to the nearest tenth, angle measures to the nearest degree, and the final area to the nearest square meter. y w P Total Area = 50