Trigonometry Sine Law and Cosine Law Page Trigonometry can be used to calculate the side lengths and angle measures of triangles. Triangular shapes are used in construction to create rigid structures. Quadrilaterals, on the other hand, are not rigid structures. In the diagram below, you can see how a rectangle can be pushed to form a parallelogram. All sides remain the same length, but the interior angles change. push A rectangle, therefore, is not a rigid structure. Similarly, any polygon with four or more sides is not a rigid structure. As stated above, however, a triangle is a rigid structure. If you push a triangle, the shape c_annot change without changing the side lengths. push Due to the rigid structure of triangles, they are a common shape in construction of roof supports and tall towers. In the previous lesson, you studied different types of triangles, which included calculating the side lengths and angle measurements of right-angled triangles. In this lesson, you will learn how to find the lengths of sides and measurements of angles of triangles that are not right-angled triangles.
Trigonometry Sine Law and Cosine Law Page 2 Non-right triangles are triangles that do not have a 90 angle. The trigonometric ratios of sin, cos, and tan are used to find the unknown measurements in right triangles only To solve for the unknown measurements in triangles that do not have a right angle, you need different formulas. One of the foil tulas you can use is the Sine Law, also known as the Law of Sines. It may be helpful for you to add this formula to your resource sheet. You should also include the situations in which the Sine Law can be used. The Sine Law in words is: "The ratio of the sine of any angle over its opposite side is a constant." The Sine Law is written as follows: sin ZA sin ZB sin ZC a O r a sin LA sin LB - sin ZC As long as you know both an angle and its opposite side, the Sine Law can help you. The Sine Law can be used in two situations. Situation One Given two angles of a triangle and one of its sides, you can solve a non-right triangle by using these steps: 1. Find the third angle by subtracting the sum of the two given angles from 1800. 2. Find the two missing sides using the Sine Law. You may remember that solving a triangle means finding the lengths of ALL the missing sides, and the measurements of ALL the missing angles. -2-
Trigonometry Sine Law and Cosine Law Page 3 Example 1 Find the lengths of side a. A t't Example 2 Solve the triangle by finding the lengths of sides a and b, and the measurement of LB. A -3-
Trigonometry Sine Law and Cosine Law Page 4 Situation Two If you are given the lengths of two sides and the measurement of one angle of a triangle, you can use the Sine Law, but only if the angle you are given is opposite one of the given sides. Example 3 Explain in which of the following triangles you could use the Sine Law. a) Example 4 Find the measurements of LA and LC of AABC, given the diagram below. -4-
Trigonometry Sine Law and Cosine Law Page 5 Example 5 Suppose you are the pilot of a commercial airliner. You find it necessary to detour around a group of thundershowers. You turn at an angle of 21 to your original path, fly for a while, turn, and intercept your original path at an angle of 35, 70 kilometres from where you left it. How much further did you have to go because of the detour? r 947,11' 3" N.,7 Example 6 70 Two observers on boats have located a sunken ship using sonar equipment. The sunken ship is in line between the two observers. The angle of depression from Observer A to the sunken ship is 40, and the angle of depression from Observer B to the sunken ship is 36. The distance between the observers is 450 metres. How far is the ship below the surface of the water? Round your answer to the nearest metre. B (water surface) N;P /01 S (sunken ship) ve Note that the depth is the perpendicular distance from the sunken ship to the surface of the water. 1. LA, LB, and LC are all angles in an isosceles triangle. If LA = 40 and LB and LC are base angles in the triangle, what is the measure of LC? 2. Which polygon has all equal sides and all interior angles equal to 90? 3. LA and LB are opposite interior angles in a parallelogram. If LA = 38, what is the measure of LB?
Trigonometry Sine Law and Cosine Law Page 6 1. Find the length of side b. a) b) 7-/ 2. Find the measurement of angle C. a) 3. Find the length of AR A * S if\ c -t -6-
Trigonometry Sine Law and Cosine Law Page 7 4. All and Beth are both looking at a hot air balloon. For Alf the angle of elevation is 40, and for Beth the angle of elevation is 58. How high is the balloon above ground if the distance between All and Beth is 600 metres? In the diagram, A represents All, B represents Beth, H represents the hot air balloon, and HC is the distance of the balloon above the ground. 1 Yu. k-2-600 in 5. Fire ranger Jarnal is 20 km due west of fire ranger Raj. They spot a fire and report the sightings as follows: m Jamal reports that the fire is N 47 0 E of his station. m Raj reports that the fire is N 20 W of his station. How far is the fire from Jamal's station? > 8 Write the values of LAJR and LARJ into the diagram before you solve the problem. -7-
Trigonometry Sine Law and Cosine Law Page 8 The Cosine Law can be used to solve certain non-right triangles. just as the Sine Law could only be used in certain situations, it is also true that the Cosine Law can only be used in certain situations. Make sure you add this formula to your resource sheet, along with the situations in which you can use the Cosine Law. The Cosine Law is written as follows for any triangle, ABC: a2 = b2 + C2-2bc cos LA b2 (12 c2 _ 7a c cos GB c2 = a2 + b2-2ab cos GC Note that there axe three versions of this formula, but there is a pattern. Notice the side squared on the left of the formula always corresponds to the angle on the right side of the formula. As with the Sine Law, in order for the Cosine Law to work, you must label your triangles correctly, where a is opposite GA, b is opposite GB, and so on Just like the Sine Law, the Cosine Law can be used in only two situations. Situation One If you are given two sides and the angle between them you can solve the triangle using the following steps. 1. Calculate the length of the missing side using the Cosine Law. 2. If required, you can use the Sine Law to calculate the measure of the smallest of the unknown angles. The Sine Law may be used more accurately with smaller angles. 3. If required, you can find the third angle using the fact that the sum of angles in a triangle is 180'. -8-
Trigonometry Sine Law and Cosine Law Page 9 Example 1 Given the triangle measurements as shown, find the measure of side a. Example 2 Find the length of side :v. Situation Two If you are given three sides of a non-right triangle, you can solve the triangle using the following steps: 1. Determine the measure of one of the larger angles using the Cosine Law (larger angles are opposite longer sides). 2, Calculate the measure of the smallest angle using the Sine Law. 3. Find the third angle using the fact that the sum of angles in a triangle is 180. L, _ ct l H kal1/41( -9-
Trigonometry -Sine Law and Cosine Law t it Page 10 USC Example 1 Find the measure of the smallest angle in triangle ABC, if a = 7, b = 8, and c =5. C. 7 :26 2- /1-, * Example 2 Solve for angles LA, LB, and LC in the following diagram. 2- t-, ( a - '2-66.7 51-10-
Trigonometry Sine Law and Cosine Law Page 11 Which formula should you use to solve the following problems the Sine Law or the Cosine Law? Do not solve the problems. a) Find the length of zu. LI) Find the length of t. c) Find the measure of LA. d) Find the measure of /D. 15-11-
Trigonometry Sine Law and Cosine Law Page 12 2. Find the measure of the side or angle indicated. a) Find side b. b) Find angle A. A c) Find side a. Jot-. AB t 1 (,( 101.1 C-- 128.7 3. The arms of a "jaws" machine used for prying open crushed vehicles in accidents are 75 cm long. The jaws are joined at one end and open at various angles. If the angle formed by the arms is 55.6, how far apart are the tips? -12-
Trigonometry Sine Law and Cosine Law Page 13 The posts of a hockey goal are 2.0 m apart. A player attempts to score by shooting the puck along the ice from a point 6.5 m from one post and 8.0 m from the other post. Within what angle, 9, must the shot be made? (Give your answer to the nearest degree.) -13-