Lesson 1: Incredibly Useful Ratios Opening Exercise Unit 6 Introduction to Trigonometry Use right triangle ΔABC to answer 1 3. 1. Name the side of the triangle opposite A in two different ways. 2. Name the side of the triangle opposite B in two different ways. 3. Name the side of the triangle opposite C in two different ways. Vocabulary When working with an acute angle within a right triangle, we identify the sides of the triangle based on its relationship to that marked angle. The side across from the right angle: The side across from the marked angle: The side next to the marked angle (not the hypotenuse): 1
Exercise For each triangle, label the appropriate sides as hypotenuse, opposite, and adjacent with respect to the marked acute angle. Vocabulary The trigonometric ratios are functions of an angle, commonly represented with the Greek letter θ (pronounced theta ). They relate the angles of a triangle to the lengths of its sides. The 3 basic trig. ratios are sine, cosine and tangent and have the following ratios: sinθ = cosθ = tanθ = Using the triangle pictured, identify the following: sin A = cos A = tan A = 2
Example a. Label the sides of each triangle with respect to the circled angle as hyp, opp, and adj. b. Find the following trig. ratios: sin A = sin B = cos A = cosb = tan A = tan B = 3
Exercises 1. Using the triangle pictured below, identify the following trig. ratios: a. sin A b. cos A c. tan A d. sin B e. cos B f. tan B 2. Using the triangle pictured below, identify the following trig. ratios: a. sin D b. cos D c. tan D d. sin E e. cos E f. tan E 4
Homework 1. Using the triangle pictured below, identify the following trig. ratios: a. sin D b. cos D c. tan D d. sin E e. cos E f. tan E 2. Looking at your answers from question 1, what do you notice about sin D and cos E? What about cos D and sin E? Do you think this will happen with ALL right triangles? Explain. 5
Lesson 2: Sine and Cosine of Complementary Angles Opening Exercise Using ΔPQR, complete the following table: θ sinθ cosθ tanθ P Q Describe any patterns you notice in the chart. Example 1 Using ΔABC, complete the following table (do not simplify the ratios): θ sinθ cosθ tanθ A B Do the patterns found in the Opening Exercise still work? 6
Example 2 Consider the right triangle ABC where C is a right angle. Find the sum of A+ B. Justify your answer. Important Discovery! The acute angles in a right triangle are always complementary. The sine of any acute angle is equal to the cosine of its complement. A+ B= 90 iff sin A= cosb Using the 2 equations listed above, how can we rewrite sin A = cosb in terms of one angle? 7
Exercises 1. Find the value that makes each statement true: a. sinθ = cos 32 b. sin 42 = cos x c. cosθ = sin ( θ + 20 ) d. cos x= sin ( x 30 ) 2. In right triangle ABC with the right angle at C, sin A = 2x + 0.1 and cosb = 4x 0.7. Determine and state the value of x. How could this relate back to the right triangle? 3. Explain why cos x = sin(90 x) for x such that 0 < x < 90. 8
Homework 1. If angles P and Q are the two complementary angles of a right triangle, complete the table: θ sinθ cosθ tanθ P sinp = 11 157 cosp = 6 157 tanp = 11 6 Q 2. Find the value that makes each statement true: a. cos12 = sinθ b. sin 2x= cos x c. sin x= cos ( x+ 38 ) d. sinθ = cos ( 3θ + 10 ) 3. In right triangle ABC with right angle at C, cos A = 4x +.07 and sin B = 2x +.13. Determine and state the value of x. Explain your answer. 9
Lesson 3: Using Trig to Find Missing Sides Opening Exercise a. Using the given diagrams, find the ratios for sind and sin A. b. Reduce these ratios. What do you notice? c. Would this also work for cosine and tangent? Why or why not? 10
We have been working with trigonometric ratios for sine, cosine, and tangent. We will now use our calculators to find the values of sin θ, cos θ, and tan θ. Your graphing calculator must be in DEGREE MODE to be able to calculate the trig values since they are all degree measures. Discussion Unit Circle! Example 1 Use a calculator to find the sine and cosine of θ. Round your answer to the nearest ten- thousandth. θ 0 10 20 30 40 50 60 70 80 90 sin θ cos θ 11
Example 2 Consider the given triangle. a. Using trig ratios, find the length of side a to the nearest hundredth. b. Now calculate the length of side b to the nearest hundredth. c. What method, other than trig, could be used to determine the length of side b? 12
Exercises Find the value of x to the nearest hundredth: 1. 2. 3. 4. 5. An 8- foot rope is tied from the top of a pole to a stake in the ground, as shown in the diagram pictured. If the rope forms a 57 angle with the ground, what is the height of the pole, to the nearest tenth of a foot? 13
Homework In 1-4, find the value of x to the nearest hundredth: 1. 2. 3. 4. 5. A hot- air balloon is tied to the ground with two taut (straight) ropes, as shown in the diagram below. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50 with the ground. a. Determine the height, to the nearest foot, of the balloon directly above the ground. b. Determine the distance, to the nearest foot, on the ground between the two ropes. 14
Lesson 4: Using Trig to Find Missing Angles Opening Exercise Think about how you solve the following equations: a. 2x = 14 b. 2 x = 9 c. How do you think we would solve 1 sin x =? 2 15
In trigonometry, to solve 1 sin x = we need to do the inverse of sin, which is arcsin. 2 Recall from your table of trig values that 1 sin x = 2 1 arcsin(sin x) = arcsin 2 1 arcsin(sin x) = arcsin 2 x = 30 To solve in your calculator: o o check that your mode is in DEGREES turn the equation into Be sure to show this work on your paper! o which is the same thing as o press and your calculator will display o type in as using the division key o hit enter to see the angle measure that has a sine value of 16
Example 1 Find the angle measure from the boy to the top of the tree. Round your answer to the nearest hundredth. 28 40 Example 2 Find the measure of the labeled angles to the nearest degree. 17
Example 4 A 16 foot ladder leans against a wall. The foot of the ladder is 7 feet from the wall. a. Find the vertical distance from the ground to the point where the top of the ladder touches the wall. Round your answer to the nearest tenth. b. Determine the measure of the angle formed by the ladder and the ground. Round your answer to the nearest degree. 18
Exercises 1. Find the measure of c to the nearest degree. 2. Find the measure of d to the nearest degree. 3. A roller coaster travels 80 ft of track from the loading zone before reaching its peak. The horizontal distance between the loading zone and the base of the peak is 50 ft. At what angle, to the nearest tenth of a degree, is the roller coaster rising? 19
Homework 1. In right triangle ABC, inches, inches, and. Find the number of degrees in the measure of angle BAC, to the nearest degree. 2. A communications company is building a 30- foot antenna to carry cell phone transmissions. As shown in the diagram below, a 50- foot wire from the top of the antenna to the ground is used to stabilize the antenna. Find, to the nearest hundredth of a degree, the measure of the angle that the wire makes with the ground. 3. As seen in the accompanying diagram, a person can travel from New York City to Buffalo by going north 170 miles to Albany and then west 280 miles to Buffalo. If an engineer wants to design a highway to connect New York City directly to Buffalo, at what angle, x, would she need to build the highway? Find the angle to the nearest degree. 20
Lesson 5: Angles of Elevation and Depression Opening Exercise From a point 120 meters away from a building, Serena measures the angle between the ground and the top of a building and finds that it measures 41. What is the height of the building? Round to the nearest meter. x Vocabulary Another way to describe the location of angles in real- world right triangle problems is the angle of elevation (when looking up at an object) and the angle of depression (when looking down at an object). Both are the angle found between the horizontal line of sight and the segment connecting the two objects. What do you think is true about the relationship between the angle of depression and the angle of elevation? Explain. 21
Example 1 A man standing on level ground is 1000 feet away from the base of a 350- foot- tall building. Find, to the nearest degree, the measure of the angle of elevation to the top of the building from the point on the ground where the man is standing. Example 2 A person measures the angle of depression from the top of a wall to a point on the ground. The point is located on level ground 62 feet from the base of the wall and the angle of depression is 52. To the nearest tenth, how far is the person from the point on the ground? Example 3 Scott, whose eye level is 1.5 m above the ground, stands 30 m from a tree. The angle of elevation of a bird at the top of the tree is 36. How far above the ground is the bird? Round your answer to the nearest tenth of a meter. 22
Example 4 Samuel is at the top of a tower and will ride down a zip line to a lower tower. The total vertical drop of the zip line is 40 ft. The zip line s angle of elevation from the lower tower is 11.5. To the nearest hundredth, how long is the zip line? Example 5 A man who is 5 feet 8 inches tall casts a shadow of 8 feet 6 inches. Assuming that the man is standing perpendicular to the ground, what is the angle of elevation from the end of the shadow to the top of the man s head, to the nearest tenth of a degree. 23
Homework 1. The Occupational Safety and Health Administration (OSHA) provides standards for safety at the workplace. A ladder is leaned against a vertical wall according to OSHA standards at an angle of elevation of approximately 75. a. If the ladder is 25 ft. long, what is the distance from the base of the ladder to the base of the wall? Round your answer to the nearest tenth. b. To the nearest tenth, how high on the wall does the ladder make contact? 2. Standing on the gallery of a lighthouse (the deck at the top of a lighthouse), a person spots a ship at an angle of depression of 20. The lighthouse is 28 m tall and sits on a cliff 45 m tall as measured from sea level. What is the horizontal distance between the lighthouse and the ship? Round your answer to the nearest whole meter. 24
Lesson 6: The Law of Sines Opening Exercise Given triangle DEF, D = 22, F = 91, DF = 16.55, and EF = 6.74, find DE to the nearest hundredth. WAIT! Can we answer this question? Why can t we use basic trig or Pythagorean Theorem to find DE? 25
The Law of Sines Used when working with 2 sides and 2 angles of ANY triangle! (Not just right triangles) Only use 2 of the fractions. a sin A = b sin B = c sinc Example 1 Let s try the Opening Exercise again using the Law of Sines! Given triangle DEF, D = 22, F = 91, DF = 16.55, and EF = 6.74, find DE to the nearest hundredth. 26
Exercises 1. In triangle ABC, m A = 33, a = 12, and m B = 43. What is the length of side b to the nearest hundredth? 2. In right Δ PQR, m Q = 90, m P = 57, p = 9.3. Find the measure of side q, to the nearest tenth. 3. In Δ XYZ, m Y = 87, y = 14 and z = 12. Find m Z, to the nearest tenth. 27
Example 2 A fire breaks out only 80 ft from the Ranger s Tower. The ranger needs to run to the water tower to open the tower spout so that it will flood the area and put out the fire. How far is the run from his tower to the water tower to the nearest foot? Water Tower 28 B Ranger's Tower C 46 80 ft A Example 3 Two lighthouses that are 30 miles apart on each side of the shorelines run north and south, as shown. Each lighthouse person spots a boat in the distance. One lighthouse notices that the location of the boat as 40 east of south and the other lighthouse marks the boat as 32 west of south. What is the distance from the boat to each of the lighthouses at the time it was spotted? Round your answer to the nearest mile. 28
Homework 1. In ΔBUG, m B =78, m G = 42, and g = 99. Find the length of side b, to the nearest tenth. 2. In Δ XYZ, m Y = 125, m X = 18,! y = 112. Find the length of side z, to the nearest hundredth. 3. A baseball fan is sitting directly behind home plate in the last row of the upper deck. The angle of depression to home plate is 30 and the angle of depression to the pitcher s mound is 24. The distance between the pitcher s mound and home plate is 60.5 feet. How far, to the nearest foot, is the fan from home plate? 29
Lesson 7: Trig. Applications Opening Exercise An archaeological team is excavating artifacts from a sunken merchant vessel on the ocean floor. To assist the team, a robotic probe is used remotely. The probe travels approximately 3,900 meters at an angle of depression of 67.4 degrees from the team s ship on the ocean surface down to the sunken vessel on the ocean floor. The figure shows a representation of the team s ship and the probe. How many meters below the surface of the ocean will the probe be when it reaches the ocean floor? Give your answer to the nearest hundred meters. 30
Example 1 As shown below, a canoe is approaching a lighthouse on the coastline of a lake. The front of the canoe is 1.5 feet above the water and an observer in the lighthouse is 112 feet above the water. At 5:00, the observer in the lighthouse measured the angle of depression to the front of the canoe to be 6. Five minutes later, the observer measured and saw the angle of depression to the front of the canoe had increased by 49. Determine and state, to the nearest foot per minute, the average speed at which the canoe traveled toward the lighthouse. 31
Example 2 The map below shows the three tallest mountain peaks in New York State: Mount Marcy, Algonquin Peak, and Mount Haystack. Mount Haystack, the shortest peak, is 4960 feet tall. Surveyors have determined the horizontal distance between Mount Haystack and Mount Marcy is 6336 feet and the horizontal distance between Mount Marcy and Algonquin Peak is 20,493 feet. The angle of depression from the peak of Mount Marcy to the peak of Mount Haystack is 3.47 degrees. The angle of elevation from the peak of Algonquin Peak to the peak of Mount Marcy is 0.64 degrees. What are the heights, to the nearest foot, of Mount Marcy and Algonquin Peak? Justify your answer. 32
Homework 1. Tim is designing a roof truss in the shape of an isosceles triangle. The design shows the base angles of the truss to have measures of 18.5. If the horizontal base of the roof truss is 36 ft. across, what is the approximate length of one side of the roof? Round your answer to the nearest hundredth. 2. A radio tower is anchored by long cables called guy wires shown as AB and AD in the given diagram. Point A is 250 m from the base of the tower. If the angle of elevation from point A to point D is 71 and the angle of elevation to point B is 65, find to the nearest meter the distance between points B and D. 33
Lesson 8: Special Right Triangles Opening Exercise There are certain special angles where it is possible to give the exact value of sine and cosine. These frequently seen angles are 0, 30, 45, 60, and 90. Using the given triangles, complete the following table and rationalize the denominators if necessary. θ 0 30 45 60 90 Sine 0 1 Cosine 1 0 Find two values in the table that are the same. What do you notice about the angle measures? Find a different set of values in the table that are the same. What do you notice about their angle measures? 34
Ratio of Sides of Special Right Triangles 30 60 90 triangle 45 45 90 triangle 2 : 2 3 : 4 2 : 2 : 2 2 3 : : 3 : : 4 : : 4 : : x : : x : : Example 1 Find the exact value of the missing side lengths in the given triangle. Example 2 Find the exact value of the missing side lengths in the given triangle. 35
Example 3 Find the exact value of the missing side lengths in the given triangle. Exercises In 1-4, find the exact value of the missing sides using special right triangles. 1. 2. 3. 4. 36
Homework In 1-4, find the exact value of the missing sides using special right triangles. 1. 2. 3. 4. 5. Given an equilateral triangle with sides of length 9, find the length of the altitude using special right triangles. Confirm your answer with the Pythagorean Theorem. 37
Lesson 9: Trigonometry and the Pythagorean Theorem Opening Exercise 1 In a right triangle with an acute angle of measure θ, sinθ=. 2 a. Draw the triangle and label the angle and known side lengths. b. Find the exact length of the missing side of the triangle. c. Find the exact value for cosθ. d. Find the exact value for tanθ. 38
Example 1 7 In a right triangle with an acute angle of measure θ, sinθ=. Using the same process as 9 the Opening Exercise, what is the value of tanθ in simplest radical form? Example 2 In lesson 3 we discovered that (x, y) = (cosθ,sinθ). Using this information, we are going to discover two more trig identities! Quotient Identity Pythagorean Identity 39
Example 3 We are going to redo the problems from the Opening Exercise and Example 1 using the Pythagorean Identity! From the Opening Exercise: 1 In a right triangle, with acute angle of measure θ, sinθ=. Use the Pythagorean Identity 2 to determine the exact value of cosθ and then use the Quotient Identity to find tanθ. From Example 1: 7 In a right triangle, with acute angle of measure θ, sinθ=. Use the Pythagorean Identity 9 to determine the exact value of cosθ and then use the Quotient Identity to find tanθ. 40
Homework 1. If 4 cosθ=, find sinθ and tanθ. 5 2. If 5 sinθ=, find cosθ and tanθ. 5 3. If tanθ= 5, find sinθ and cosθ. 41