Unit 3 Trigonometry. 3.1 Use Trigonometry to Find Lengths

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1 Topic : Goal : Unit 3 Trigonometry trigonometry I can use the primary trig ratios to find the lengths of sides in a right triangle 3.1 Use Trigonometry to Find Lengths In any right triangle, we name the sides by their relationship with one of the non right angles. In the following triangle, the non right angle we have chosen has been labelled with a. opposite adjacent hypotenuse For any right triangle that has an angle the same as the ratios of side lengths will always be the same and are defined as follows... opposite sin = hypotenuse tan = opposite adjacent adjacent cos = hypotenuse When you use the trig ratios you must always choose an ANGLE first. It will either be the angle you intend to use to find a side, or the angle you are to find the value of. Today we will be using trigonometry to find the sides in a triangle. Example 1. Finding side when the unknown is in the NUMERATOR of the ratio. Find the indicated side length. x 25 cm First circle the angle you are using, then label the sides as opposite, adjacent or hypotenuse. Determine which ratio you need to use and set up the ratio. 27 o

2 Example 2. Finding the length of side when the unknown is in the denominator of the ratio. 33o x 15 cm Example 3. The angle of depression from a helicopter to a forest fire is 15o. If the helicopter is flying towards the fire at an altitude of 900m, how far is it from being directly over the fire? Homework Page 189 #1, 2, 4, 6, 7, 8, 10, 13

3 Topic : Goal : trigonometry I can find the angles in a non right triangle using the primary trig ratios. 3.2 Use Trigonometry to Find Angles If you know two sides in a right triangle, you can determine either of the non right angles by setting up ratios. Just remember SOH CAH TOA i n e To tell the calculator that you are inputting the ratio and you want it to give you the angle, you need to use the inverse sine, cosine and tangent buttons. It looks like sin 1. You need to press 2nd, Inv or Shift key (whichever you have on your calculator). o s i n e a n g e n t Example 1. Find the indicated angle in the following triangles

4 Example 2. A ramp is built up to the door of a house. By safety standards, the angle of elevation of the ramp must be no more than 10 o. If the door is 56 cm above the ground and the ramp is 1.7m long, does it meet safety standards? Homework Page 194 #1, 2 4ac, 6 11

5 Topic : Goal : Right angle Trigonometry I can solve for missing information in questions that involve more than one right triangle. 3.3 Solve Problems Involving 2 Right Triangles Sometimes in order to find the the sides or angles asked for, we have to find other sides or angles in intermediate steps. We'll go through some examples that involve this. Usually angles are the easiest to find REMEMBER that the sum of the angles in ANY triangle is 180 o. If you are at a loss of where to begin, start by finding all the angles that you can. Example 1. Find the indicated side length. D A 32 ft x HINT Remember that straight lines also have 180 o 118 o B 27 ft C 41 ft E

6 HINT Example 2. Find the length of side BD. A If the triangles share a common side, you will likely need to find it's measure 14m 12 O 67m 28 O B C D Example 3. As the sun is setting, the angle of elevation to the sun (from ground level) goes from 32 O to 27 O. During this time the shadow of a tree with lengthen. By how much does the shadow lengthen if the tree is 18ft tall? Homework Page 200 #1 6, 8

7 Today's Topic : Today's Goal : Sine Law for Non Right Triangles I know how to solve for sides and angles in nonright triangles using The Law of Sine's. 3.4 The Sine Law Unfortunately not all triangles are right triangles. The primary trig rations can only be used when you have a right triangle. But we can use our primary trig rations to develop some formulas that will work for ALL triangles. Proof: Given ABC draw AD BC, where AD is the height of the triangle. A B C

8 If we had drawn the altitude to a different side, we would go through the same process to see that side a and angle A are involved in the same ratio. So we have developed sine law. A brief warning sine law doesn t work for finding obtuse angles (those greater than 90 o ) You should never use sine law to find the angle opposite the largest side of a triangle, just in case it is obtuse. Example Using Sine Law to Solve Right Triangles (remember to solve means to find all missing measures) a)

9 b) Homework Page 207 #(1,2)bd, 3, 4

10 Topic : Sine Law Goal : I can use the sine law to solve application problems. 3.5 Word Problems Using the Sine Law Example 1.

11 Example 2. A bridge across a valley is 150 m in length. The valley walls make angles of 60 and 54 with the bridge that spans it, as shown. How deep is the valley, to the nearest metre? Example 3.

12 Today's Topic : The Cosine Law Today's Goal : to understand how to tell when a situation requires the Law of Cosines, and to apply it to find sides and angles in a non right triangle. 3.6 The Cosine Law We learned that to use Sine Law, you need to have one angle and its opposite side to both have numbers on them. There are then two instances where Sine Law is of no use to us. We have all three sides with known values. We have two sides and a contained angle. In these cases we will need to introduce the Law of Cosines. If in ΔABC sides a, b, and c are opposite angles A, B and C respectively, the following property holds c 2 = a 2 + b 2 2abCosC NOTE : 'a' and 'b' are the sides that make up angle C and 'c' is the side directly across the triangle from angle C. or if we rearrange it to solve for an angle So here we have the two versions of the Cosine Law. For finding SIDES. For finding ANGLES c 2 = a 2 + b 2 2abCosC CosC = a 2 + b 2 c 2 2ab Example 1. Find the value of side c in the given triangle. c 2 = a 2 + b 2 2abCosC Always remember that whatever side you are finding is on the left hand side of the equation, and the angle across from it is the value of the far right.

13 Example 2. Solve the following triangle. When using cosine law to solve a triangle, you will only have to use it once, and then you will have an angle across from its opposite side, and therefore you can use the sine law. But remember you should never use sine law to find the biggest angle in a triangle in case it is obtuse. So the first angle we should find is the one across from the biggest side. If there is an obtuse angle in the triangle, that will be it. Example 3. A plane leaves London and travels N55 o E for 350 km. Another plane leaves London travelling at S85 o W, and travels for 460 km. How far apart are the planes. N

14 Topic : problems with triangles Goals : I know when to use Sine Law and Cosine law if I'm not told which to use in the the question. 3.7 Making Connections with Sine Law and Cosine Law a b sina= sinb need two sides and the angle between cosc = a 2 + b 2 c 2 2ab need all three sides sina a = sinb b need angle across from side and another side c 2 = a 2 + b 2 2abcosC need angle across from side and another angle Example 1. To get around an obstacle, a local utility must lay two sections of underground cable that are 371.0m and 440.0m long. The two sections meet at an angle of How much EXTRA cable is needed to go around the obstacle as opposed to a straight line distance between? Example 2. After the ice storm, a neighbour needed help to stabilize a small tree in his front yard that was now leaning. To prop it up, we used a 6ft piece of board. we attached it to the ground 5 ft from the base of the trunk and attached it to the tree, 4.3 ft up the length of the trunk. Through was angle was the tree leaning?

15 Example 3. A poll tilts towards the sun at an 8 o angle from the vertical and it casts a 22 ft shadow. The angle of elevation from the shadow to the top of the pole is 43. How tall is the poll? Example 4. Two planes leave an airport at the same time. One plane is flying 575 m.p.h at a bearing N 23 o E, and the other plane is flying at 625 m.p.h at a bearing of N 65 o W. How far apart are the planes after flying for 3.5 hours? Homework Page 220 #(1 3) 4, 5, 8, 11, 13, 18, just decide which each needs and solve one of each.