Name: Trigonometry Ratios A) An Activity with Similar Triangles Date: For each of the right triangles below, the labelled angle is equal to 40. Why then are these triangles similar to each other? Page 1 of 13
Complete the table below by measuring (using a ruler) the side length (to the nearest millimeter) of each triangle (on the previous page). Triangle Side Opposite to 40 o Angle Side Adjacent to 40 o Angle Side Name Measurement Side Name Measurement ABC BC AC Ratio Opposite Adjacent BC AC = Ratio Decimal Form 4 places DEF GHJ KLM NOP How are the ratios in the last column related? If the measure of one angle in a right angled triangle is kept constant, then what can you conclude about the following ratio: length of the side opposite the angle length of the side adjacent to the angle Page 2 of 13
B) The Tangent Ratio Trigonometry is the study of the ratios of the lengths of sides of triangles. It is based on the property of similar triangles. The results of the activities on pages 1 2 indicate the following conclusion: For all right triangles with a given angle, the ratio below is a constant and we define this ratio as tangent. Tangent θ = length of the side opposite toθ length of the side adjacent toθ In normal use, we shorten this to: OPP ADJ Using a proper Scientific Calculator: If you do not have a proper scientific calculator you need to get one today! Your calculator will likely have three modes for trig (degrees, radians and gradients). Learn how to switch between the three modes and make sure your calculator is always in DEGREE mode when completing exercises! To determine tan 32, follow these steps. Your calculator may do method 1 or it may do method 2. Try both to see which your calculator does and then use that method with your calculator from now on. If you borrow a calculator, it may use the other method, so be sure to know both methods, just in case. Method 1 Punch: 32 TAN Your display should be 0.624869 Method 2 Punch: TAN 32 = Your display should be 0.624869 Examples: Find each of the following tangents. Round to 4 decimal places. a) tan 54 = 1.3768 b) tan 60 = 1.7321 c) tan 75 = d) tan 14 = e) tan 45 = f) tan 30 = Answers: c) 3.7321 d) 0.2493 e) 1 f) 0.5774 Page 3 of 13
We can also use the calculator to find the angle, knowing the tangent, as follows: To determine the angle, θ, whose tangent is 0.8391 to the nearest degree. Method 1: Punch: 0.8391 2 nd TAN or 0.8391 SHIFT TAN Your display should read 40.0000 or rounded correctly as 40 o. Method 2: Punch: 0.8391 2 nd TAN = or 0.8391 SHIFT TAN = Your display should read 40.0000 or rounded correctly as 40 o. Note: By punching 2 nd or SHIFT TAN you are inputting tan -1 in your calculator. tan -1 is called the inverse tangent. We read tan -1 as tan inverse. Examples: Find the angles whose tangents are shown below. Round your angle measurements to the nearest degree. a) 3.4874 b) 0.1234 c) 2.3456 d) 0.7890 θ = tan -1 (3.4874) θ = 74 Answers: b) 7 c) 67 d) 38 Page 4 of 13
C) Using the Tangent Ratio We can use the tangent of an angle to find the length of a side of a right triangle. For example, to find x (to 1 decimal place) in the triangle shown: There is no need for an introductory statement since x is already labeled on the diagram. OPP ADJ x tan40 = 21 x = 21tan40 x = 17.6 There is no need to show the value of tan 40. Do the whole calculation in one step on the calculator. We can also use the tangent to find an angle of a right triangle. For example, find θ, to the nearest degree, in the triangle shown: OPP ADJ 10 13 1 10 θ = tan 13 θ = 38 Use brackets around the fraction and evaluate in one step. Page 5 of 13
Another example: Find x, to the nearest unit, in the triangle shown below OPP ADJ 17 tan34 = x 17 x = tan34 x = 25 Use your calculator in the final step. You should keep all decimal places in your calculations. Only round your final answer. We can even use the tangent ratio to solve word problems. For example: John places a ladder on level ground 3 m from a vertical wall so that the ladder makes a 70 angle with the ground. How far up the wall does the ladder reach? Introductory Statement: Let x represent the distance the ladder reaches up the wall (in metres). Draw a diagram. OPP ADJ x tan70 = 3 x = 3tan70 x = 8.242432258 x = 8.24 m Concluding Statement: The ladder reaches 8.24 m up the wall. Note: If the question does not clearly state how your answer should be expressed, as a rule of thumb, we round the answer correct to the nearest metre because the distance from the wall was given to the nearest metre. Page 6 of 13
D) Introduction to the Sine and Cosine Ratios Use the triangles on page 1 from Part A (An Activity with Similar Triangles) in this ISU (in which θ = 40 ) to complete the following table. Triangle Side Opposite to 40 o Angle Side Adjacent to 40 o Angle Hypotenuse Sine Ratio Opposite Hypotenuse Cosine Ratio Adjacent Hypotenuse Decimal Form 4 places Decimal Form 4 places ABC DEF GHJ KLM NOP How are the ratios in the Sine column related? How are the ratios in the Cosine column related? Page 7 of 13
You now have the three basic trigonometric ratios for any acute angle: In a right angled triangle: Long form: sinθ = cosθ = opposite hypotenuse adjacent hypotenuse opposite adjacent Short form: opp sinθ = hyp cosθ = adj hyp opp adj Memory Aid Page 8 of 13
Practice: Use the sine ratio to find the value of x in each of the following: (Round answers to the nearest unit). opp sinθ = hyp x sin40 = 20 x = 20sin40 x = 13 b) c) Answers: b) 19 c) 97 Page 9 of 13
E) Angles of Elevation and Depression The angle of elevation is used when looking up at an object. It is the angle between looking straight ahead and then lifting your eyes to the object. The angle of depression is used when looking down at an object. It is the angle between looking straight ahead and then lowering your eyes to the object. Note: Angles of elevation and depression are always measured relative to the horizontal, NEVER relative to the vertical. Example: The angle of elevation to the top of a building is 72 from a point 60 m from the foot of the building. Find the height of the building to the nearest meter. opp adj h tan72 = 60 h = 60tan72 h = 184.661 The height of the building is 185 m. Page 10 of 13
Homework: Tangent Ratio: 1. Determine x in each of the following triangles. Round your answers to the nearest whole number. a) b) c) 2. Determine the height of a tree casting a 20 m shadow at the same time of day as the sun s rays make an angle of 35 with the ground. 3. How tall is a flagpole, to the nearest centimetre if it casts a shadow 15.5 m long when the sun s rays make a 25 angle with the ground? 4. Commercial airplanes fly at about 10 km above ground. If the landing approach is to make a 5 angle with the ground, how far from the airport, in kilometres must the pilot begin the descent? 5. From the top of a cliff, the angle of depression of a hut is 46. If the cliff is 500 m high, how far is the hut from the base of the cliff in metres? 6. From the top of a cliff 120 m above the water, the angle of depression of a boat on the water is 18. How far is the boat from the base of the cliff? 7. A tower 115 m high casts a shadow 24 m long. Find the angle of elevation of the sun. 8. A 10-story building (each story is 3 m high) casts a shadow of 55 m. What is the angle of elevation of the sun? Page 11 of 13
Sine and Cosine Ratio: 1. Use the sine ratio to find the length of the side indicated. a) b) c) 2. Use the cosine ratio to find the value of y in each of the following triangles. Round your answers to the nearest unit. a) b) c) d) e) f) 3. For the following triangle, state the value of: a) sinθ b) cosθ c) tanθ d) θ Page 12 of 13
For each of the following, include a neatly labeled diagram. 4. A support cable 60 m long connects the top of a pole to the ground. The cable makes an angle of 72 with the ground. Calculate the height of the pole to the nearest metre. 5. How far from the wall (to the nearest centimetre) must the foot of a 10 m ladder be placed in order to make a 79 angle with the ground? 6. A 14 m ladder rests against the wall of a house. The foot of the ladder rests on level ground 2.5 m from the wall. What angle does the ladder make with the ground? 7. A tree known to be 50 m high casts a shadow that is 17 m long. What angle do the sun s rays make with the ground at this time of day? 8. From the top of a building, a SWAT team member spots a perpetrator at an angle of depression of 41. If the perpetrator is 25.9 m from the base of the building, find the height the building. 9. What angle will a 73 m wire make with the ground if it secures a tower from a point 45 m up the tower? 10. A tree casts a 23 m shadow when the angle of elevation of the sun is 67. a) Find the height of the tree. b) Find the length of the shadow when the angle of elevation of the sun is 37. Tangent Homework Answers 1. a) 14 b) 21 c) 39 2. The tree is 14 m tall. 3. The flagpole is 7.23 m high. 4. The airplane should start its descent 114 km from the airport. 5. The cliff is 483 m from the hut. 6. The boat is 369 m from the cliff. 7. The angle of elevation of the sun is 78. 8. The angle of elevation of the sun is 29. Sine and Cosine Ratio Homework Answers 1. a) x = 34 b) x = 81 c) x = 107 2. a) y = 9 b) y = 24 c) y = 13 d) y = 35 e) y = 18 f) y = 75 3. 12 5 12 a) sinθ = b) cosθ = c) 13 13 5 d) θ = 67.4 4. The pole is 57 m high. 5. The ladder is 1.91 m out from the wall. 6. The ladder makes an angle of 80 with the ground. 7. The angle of elevation of the sun is 71. 8. The building is 22.5 m tall. 9. The wire makes an angle of 38 with the ground. 10. a) The tree is 54 m tall. b) The new shadow is 72 m long. Page 13 of 13