Download Unit 4: Trigonometry

11 Applied Mathematics Name: ___________________________ Unit 4: Trigonometry LESSON/HOMEWORK LOG Lesson Lesson 4.1 Exploring Side-­‐ Angle Relationships in Acute Triangles(3.1) p. 117 #1-­‐5 Lesson 4.2 Proving & Applying the Sine Law (3.2) p. 124 #1-­‐4, 6, 9, 10, 12, 13 Mid-­‐ Chapter Review (Ch. 3) p. 129 #1-­‐9 Lesson 4.3 Proving & Applying the Cosine Law (3.3) p. 137 #1-­‐5, 10, 13 Lesson 4.4 Solving Problems Using Acute Triangles (3.4) p. 147 #1-­‐5, 7-­‐13 ODD Review-­‐ Chapter 3 p. 152 All (optional) p. 154 All Lesson 4.5 Exploring the Primary Trigonometric Ratios of Obtuse Angles (4.1) p. 163 #1-­‐4 Lesson 4.6 Proving & Applying the Sine and Cosine p. 171 #1-­‐6, 8, 9, 11, 12, 14 Laws for Obtuse Triangles (4.2) Mid-­‐Chapter Review (Ch. 4) p. 175 All Lesson 4.7 The Ambiguous Case of the Sine Law (4.3) p. 183 #1, 4-­‐6, 9, 11, 13 Lesson 4.8 Solving Problems p. 194 #2, 3, 5, 7-­‐13 ODD Using Obtuse Triangles (4.4) Review-­‐ Chapter 4 p. 198 All p. 200 All Unit Review (Ch. 3 & 4) Handouts Hom ew ork Assignm ent How did you do? Date 1 (not well) 2 3 (not bad) 4 5 (perfection!) 11 Applied Mathematics Lesson 4.1: Exploring Side-­‐Angle Relationships in Acute Triangles Sine Law Exploration Instructions 1. Choose three consecutive letters and label triangle one. Use lowercase letters for sides and capital letters for angles. 2. Measure all of the sides and all of the angles of triangle one. Measure more than once to check for accuracy. 3. Record the confirmed measurements on your paper. 4. Set up and solve a ratio for each side/angle pairing in triangle one, like this: 𝑎
12
= = 13.85 sin 𝐴
sin(60)
5. Write in a sentence what you observe about the three ratios. 6. Check if this observation is true for all triangles by repeating steps 1-­‐4 for triangle two. Triangle One Observation: ________________________________________________________________________________________ _______________________________________________________________________________________________________ Triangle Two Was your observation confirmed? _____________ 2 11 Applied Mathematics Key Idea: The ratios of 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆
𝒔𝒊𝒏 (𝒂𝒏𝒈𝒍𝒆)
in an acute triangle. The Sine Law: In an acute triangle, ∆ABC, are equivalent for all three side-­‐angle pairs 𝒂
𝒔𝒊𝒏 𝑨
𝒔𝒊𝒏 𝑨
Or 𝒂
=
𝒃
𝒔𝒊𝒏 𝑩
=
𝒄
𝒔𝒊𝒏 𝑪
= 𝒔𝒊𝒏 𝑩
= 𝒔𝒊𝒏 𝑪
𝒃
𝒄
NOTE: When labeling triangles, -­‐ Angles -­‐ Side lengths Capitol and lower case letters correspond; the same letter is used for an angle and its opposite side. The longest side is opposite the largest angle. The shortest side is opposite the smallest angle. The middle side is opposite the middle measure angle. Lesson 4.2: Proving & Applying the Sine Law In the previous lesson, we discovered the side-­‐angle relationship in acute triangles. We want to prove that it works in all acute triangles. Consider the proof given on p. 118 of the textbook. Answer the questions A-­‐I. 3 11 Applied Mathematics Example 1: A triangle has angles measuring 75O and 60O. The side opposite the 75O angle is 14.0 m in length. Determine the length of the side opposite the 60O, to the nearest tenth of a metre. Example 2: Toby uses chains attached to hooks on the ceiling and a winch to lift engines at his father’s garage. The chains, the winch, and the ceiling are arranged as shown. Toby solved the triangle using the sine law to determine the angle that each chain makes with the ceiling to the nearest degree. He claims θ = 40O and α = 54O. Is he correct? Explain and make any corrections. 4 11 Applied Mathematics Example 3: The captain of a small boat is delivering supplies to two lighthouses, as shown. His compass indicates that the lighthouse to his left is located at N30OW and the lighthouse to his right is located at N50OE. Determine the compass direction he must follow when he leaves lighthouse B for lighthouse A. NO TE: You can
use t he si ne l aw w hen you kn ow: • tw o si des and t he an gle o ppos i te a kno wn s ide (S S A) • tw o ang les and a ny si de (SA A or AS A) • If you kn ow th e m eas ure s of tw o ang les in a trian gl e, you can dete rm i ne t he thi rd an gle , becaus e the a ngl es m us t ad d to 18 0 O 5 11 Applied Mathematics Lesson 4.3: Proving & Applying the Cosine Law The sine law cannot always help you determine the unknown side lengths and angle measures. Consider these triangles below. Try to solve using the sine law. The Cosine Law: 𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄 𝒄𝒐𝒔𝑨 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄 𝒄𝒐𝒔𝑩 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃 𝒄𝒐𝒔𝑪 Rearrange to solve for an angle: 𝒃𝟐 + 𝒄𝟐 − 𝒂𝟐
𝒄𝒐𝒔 𝑨 =
𝟐𝒃𝒄
The proof for the Cosine Law is on p. 130. 6 11 Applied Mathematics NO TE : You can use t he cos in e law w hen you kn ow: • tw o si des and t he con tai ned a ng le (SA S) •
all three s id es (S SS ) When using the cosine law, follow these steps: 1. Label the triangle so that what you want to know is a or A 2. Make a list of what you know 3. Identify the unknown 4. Write out the appropriate formula 5. Plug your known values in to the formula 6. Solve for the unknown 7. Write out your final answer with the appropriate units. Example 1: Determine the length of CB to the nearest centimetre. 7 11 Applied Mathematics Example 2: Determine the measure of ∠A to the nearest degree. Example 3: The diagram at the right shows the plan for a roof, with support beam DE parallel to AB. The local building code requires the angle formed at the peak of a roof to fall within a range of 70O to 80O so that snow and ice will not build up. Will this plan pass the local building code? 8 11 Applied Mathematics Lesson 4.4: Solving Problems Using Acute Triangles The sine law, the cosine law, the primary trigonometric ratios, and the sum of angles in a triangle may all be useful for solving acute triangle problems. When selecting the sine law or the cosine law, consider which sides and/or angles are given and which side or angle is to be determined. (See also the table on p. 146) Steps to Choose a Law: 1. 2. 3. How You Know You Have Chosen Correctly NOTE: You may have to use both laws or one law twice to answer a question. Example 1: Determine the height of AC to the nearest metre. 9 11 Applied Mathematics Example 2: A kayak leaves a dock on Lake Athabasca and heads due north for 2.3 km. At the same time, a second kayak travels in a direction N55OE from the dock for 1.8 km. In which direction, to the nearest degree, would the second kayak have to travel to meet the first kayak? 10 11 Applied Mathematics Lesson 4.5: Exploring the Primary Trigonometric Ratios of Obtuse Angles Oblique Triangle: a triangle that does not contain a angle. Read Explore the Math on p. 162. Complete the following table: θ 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽 𝐭𝐚𝐧 𝜽 𝟏𝟖𝟎𝑶 − 𝜽 𝐬𝐢𝐧 (𝟏𝟖𝟎𝑶 − 𝜽) 𝐜𝐨𝐬 (𝟏𝟖𝟎𝑶 − 𝜽) 𝐭𝐚𝐧 (𝟏𝟖𝟎𝑶 − 𝜽) 100! 110! 120! 130 180! !
What relationships do you observe when comparing the trigonometric ratios for obtuse angles with the trigonometric ratios for the related supplementary acute angles? For any angle θ , • 𝐬𝐢𝐧 𝜽 = 𝐬𝐢𝐧( 𝟏𝟖𝟎𝑶 − 𝜽) • 𝐜𝐨𝐬 𝜽 = − 𝐜𝐨𝐬 ( 𝟏𝟖𝟎𝑶 − 𝜽) • 𝐭𝐚𝐧 𝜽 = − 𝐭𝐚𝐧 (𝟏𝟖𝟎𝑶 − 𝜽) Example 1: Determine the measures of two angles between 0O and 180O that have each sine ratio. Round each answer to the nearest degree. a) 0.34 !
b) !
11 11 Applied Mathematics Lesson 4.6: Proving & Applying the Sine and Cosine Laws for Obtuse Triangles Prove that the Sine Law holds true for obtuse triangles. 12 11 Applied Mathematics Example 1: Determine the measure of ∠C to the nearest degree. Example 2: In an obtuse triangle, ∠B measures 23.0O and its opposite side, b, has a length of 40.0 cm. Side a is the longest side of the triangle, with a length of 65.0 cm. Determine the measure of ∠A to the nearest tenth of a degree. 13 11 Applied Mathematics Example 3: Colleen and Juan observed a tethered balloon advertising the opening of a new fitness centre. They were 250 m apart, joined by a line that passed directly below the balloon, and were on the same side of the balloon. Juan observed the balloon at an angle of elevation of 7O while Colleen observed the balloon at an angle of elevation of 82O. Determine the height of the balloon to the nearest metre. Example 4: Show that the Cosine Law holds for obtuse triangles. 14 11 Applied Mathematics Example 5: The roof of a house consists of two slanted sections. A roofing cap is being made to fit the crown of the roof, where the two slanted sections meet. Determine the measure of the angle needed for the roofing cap, to the nearest tenth of a degree. roofing cap 20.3 ft 17.0 ft 33.5 ft NOTE: The sine and cosine law are used with obtuse triangles in the same way that they are used with acute triangles. 15 11 Applied Mathematics Lesson 4.7: The Ambiguous Case of the Sine Law You must check for the ambiguous case of the sine law when you are given two side lengths and the measure of an angle opposite one of the sides: SSA. The ambiguous case of the sine law occurs when the available information. can be drawn, given Example 1: Given each SSA situation for ∆ABC, determine how many triangles are possible. a) ∠A = 30O, a = 4 m, b = 12 m b) ∠A = 30O, a = 6 m, b = 12 m c) ∠A = 30O, a = 8 m, b = 12 m d) ∠A = 30O, a = 15 m, b = 12 m 16 11 Applied Mathematics Example 2: Isha is at the starting point of two southeasterly paths at an angle measure of 25O to each other. She followed these directions: • Follow the more southerly path for 10.2 km to a lookout. • Turn due north and walk another 5.4 km to a cabin. • Turn and walk directly back to the starting point along the more northerly path. What is the compass direction of the return path, to the nearest degree? 17 11 Applied Mathematics Example 3: Martina and Carl are part of a team that is studying weather patterns. The team is about to launch a weather balloon to collect data. Martina’s rope is 7.8 m long and makes an angle of 36.0O with the ground. Carl’s rope is 5.9 m long. Assuming that Martina and Carl form a triangle in a vertical plane with the weather balloon, what is the distance between Martina and Carl, to the nearest tenth of a metre? 18 11 Applied Mathematics Summary of Ambiguous Case 19 11 Applied Mathematics Lesson 4.8: Solving Problems Using Obtuse Triangles When solving problems, consider the following flow chart: Draw a diagram. Label it with all the given information. Right Triangle Obtuse Triangle Acute Triangle Use the primary trigonometric ratios. Use the sine law or the cosine law. With the sine law, remember to consider the ambiguous case. Example 1: An aerial cable car takes passengers across a 74 m wide gorge. At one point in the crossing, the angles of depression of the cable supporting the cable car measure 43O and 37O. Determine the length of the cable, to the nearest tenth of a metre. 20 11 Applied Mathematics Example 2: From a 10 m lookout tower, the angles of depression to two tents measure 22O and 29O. One tent is on a heading of S35OW. The other is on a heading of S68OE. What is the distance between the two tents, to the nearest metre? 21