Review on Right Triangles Identify a Right Triangle Example 1. Is each triangle a right triangle? Explain. a) a triangle has side lengths b) a triangle has side lengths of 9 cm, 12 cm, and 15 cm of 5 cm,7 cm, and 9 cm Try. A triangle has side lengths of 12 cm, 16 cm, and 20 cm. Is the triangle a right triangle? Explain. Apply the Pythagorean Theorem Example 2. Find the length of the unknown side. Use a calculator to approximate to 1 decimal place. Try. Find the length of the unknown side. Use a calculator to approximate each answer to 1 decimal place. 35
The side across from the right angle is called the hypotenuse (hyp). It is the LONGEST side of a right-angled triangle. If you stand at A, the side BC is opposite to you and the side AB is next to you. We call that BC is the opposite side (opp) and AB is the adjacent side (adj). Name the Sides in a Right Triangle Example 3. Name the sides of each right triangle. Try. Find the trigonometric ratio of each angle to the nearest thousandth. a) cos (75 ) b) sin (25 ) c) tan (72 ) Try. Find A to the nearest degree if trigonometric ratio is known. a) tan A = 1.327 b) cos A = c) sin A = 36
Recall the definitions of the three Primary Trigonometric Ratios: The rules for determining the sine ratio, cosine ratio, and tangent ratio for an angle in a right triangle can be memorized by using the acronym SOH CAH TOA. Determine the Unknown Angle in a Right Triangle Example 4. In right ABC, find tan C, then C. Try. In right PQR, find the measures of P. Try. In right ABC, find C. 37
Determine the Unknown Side Length in a Right Triangle Example 5. Use trigonometric ratios to determine the unknown side in the given triangle. Answer correct to the nearest tenth of an unit. Example 6. Use trigonometric ratios to determine the unknown side in the given triangle. Answer correct to the nearest tenth of an unit. Try. Find the unknown side to the nearest tenth. 38
3.1 Applying the Sine Law on Acute Triangles We use primary trigonometric ratios, SOH CAH TOA, to solve right triangles. What about the triangles that are not right-angled, say acute triangles? Note: lower-case letters a, b, c represent the opposite sides from A, B and C respectively. Apply Sine Law to Find Unknown Side Length Example 1. Find each measurement indicated. Round your answers to the nearest tenth. Try. Find each measurement indicated. Round your answers to the nearest tenth. 13 39
Apply Sine Law to Find Unknown Angle Measure Example 2. Find each measurement indicated. Round your answers to the nearest whole degree. Try. Find each measurement indicated. Round your answers to the nearest whole degree. 40
3.2 Proving the Sine Law and Problem Solving on Acute Triangles Use Reasoning to Prove Sine Law Example 1. Prove sine law using deductive reasoning. 41
Use Sine Law to Solve Problem 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. Determine the angle that each chain makes with the ceiling to the nearest degree. Example 3. Allison is flying a kite. She has released the entire 150 m ball of kite string. She notices that the string forms a 70 angle with the ground. Marc is on the other side of the kite and sees the kite at an angle of elevation of 30. How far is Marc from Allison, to the nearest tenth of a metre? Directions are often stated in terms of north and south on a compass. For example, N30 E means travelling in a direction 30 east of north. S45 W means travelling in a direction 45 west of south. 42
Use Sine Law to Solve Direction Problem Example 4. Janice is sailing from Gimli on Lake Winnipeg to Grand Beach. She had planned to sail 26.0 km in the direction S71 E; however, the wind and current pushed her off course. After several hours, she discovered that she had actually been sailing S79 E. She checked her map and saw that she must sail S18 W to reach Grand Beach. Determine, to the nearest tenth of a kilometre, the distance remaining to Grand Beach. Example 5. 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 N30 W and the lighthouse to his right is located at N50 E. Determine the compass direction he must follow when he leaves lighthouse B for lighthouse A. 43
3.3 Applying the Cosine Law on Acute Triangles (Part I) Example 1. Find each measurement indicated. Round your answers to the nearest tenth. The sine law cannot always help you determine unknown angle measures or side lengths. You can use the sine law to solve a problem modelled by an acute triangle when you know: - two sides and the angle opposite a known side - two angles and any side You can use the cosine law to solve a problem that can be modelled by an acute triangle when you know: two sides and the contained angle all three sides 44
Apply Cosine Law to Find Unknown Side Length Example 2. Find each measurement indicated. Round your answers to the nearest tenth. Try. Find each measurement indicated. Round your answers to the nearest tenth. 45
Apply Cosine Law to Find Unknown Angle Measure Example 3. Find each measurement indicated. Round your answers to the nearest whole degree. Try. Find each measurement indicated. Round your answers to the nearest whole degree. 46
3.3 Proving the Cosine Law and Problem Solving on Acute Triangles (Part II) Use Reasoning to Prove Cosine Law Example 1. Prove sine law using deductive reasoning. 47
Solve a Problem Using the Cosine Law Example 2. The radar screen of a Coast Guard rescue ship shows that two boats are in the area, N45 E and S50 E. How far apart are the two boats, to the nearest tenth of a kilometre? Try. Two aircraft, A and B, leave an airport at the same time. A flies on a course of N90 E at 700 km/h, and B flies on a course of N70 W at 600 km/h. Calculate their distance apart in 12 minutes to the nearest kilometre. Example 3. Two airplanes leave the Hay River airport in the Northwest Territories at the same time. One airplane travels at 355 km/h. The other airplane travels at 450 km/h. About 2 h later, they are 800 km apart. Determine the angle between their paths, to the nearest degree. 48
3.4 Solving Problems Using Acute Triangles To decide whether you need to use the sine law or the cosine law, consider the information given about the triangle and the measurement to be determined. Solve a 2-D Problem Example 1. Ryan is in a police helicopter, 400 m directly above the Sea to Sky highway near Whistler, British Columbia. When he looks north, the angle of depression to a car accident is 65. When he looks south, the angle of depression to the approaching ambulance is 30. How far away is the ambulance from the scene of the accident, to the nearest tenth of a metre? 49
Example 2. The first hole at a golf course is 210 yards long in a direct line from the tee to the hole. Andrew Duffer hit his first shot at an angle of 15 off the direct line to the hole. The angle between his first shot and his second shot was 105. His second shot landed in the hole. What was the length of his second shot, to the nearest yard? Example 3. An oil company drilling off shore has pipelines from platform Beta to the same shore station Delta. Platform Alpha is 180 km on a bearing of 50 from Delta and platform Beta is 250 km on a bearing of 125 from Delta. Calculate the distance between platform Alpha and platform Beta to the nearest km. Example 4. From a point A, level with the foot of a hill, the angle of elevation of the top of the hill is 16. From a point B, 950 metres nearer the foot of the hill, the angle of elevation of the top is 35. What is the height of the hill, DC, to the nearest metre? 50
Example 5. P and Q are two bases for a mountain climb. PQ is 600 m and QR is a vertical stretch of a rock face. The angle of elevation of Q from P, (i.e. QPS) is 31. The angle of elevation of R from P, (i.e. RPS) is 41. Mark these measurements on the diagram and state the measures of RPQ and PRQ. Use the sine law in PQR to calculate the height of the vertical climb, QR, to the nearest metre. Solve a 3-D Problem Example 6. On June 30, 1956, the world s largest free standing totem pole was erected in Beacon Hill Park in Victoria. Recently, a surveyor took measurements to verify the height, h, of the totem pole. In the diagram, triangle ABC lies in a vertical plane and triangle BCD lies in a horizontal plane. 51