Grade 10 Academic (MPM2D) Unit 6: Trigonometry 2: Non-Right Triangles Applications of Sine & Cosine Laws Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Trigonometry is a branch of mathematics that studies the relationship between the measures of the angles and the lengths of the sides in triangles. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Notation: In any triangle, the vertices are named using capital letters as in the given example. The lengths of the sides are named using lower case letters that match the opposite vertex. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Sine (Sin), COSINE (Cos), TANGENT (Tan): Opposite side Adjacent side Hypotenuse side Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Sine Law Cosine Law a Using Cosine Law to find the angles or © 2017 E. Choi – MPM2D - All Rights Reserved Applications of Sine & Cosine Laws

Steps to solve triangle Right triangles can be solved with the primary trig. Ratios Non-rights triangles can be solved with sine law & cosine law. Use Sine Law if: - 2 sides and one opposite angle to one of the given sides are known (SSA) - 2 angles and one opposite side to one of the given angles are known (AAS) Use Cosine Law if: - all 3 sides are known (SSS) - 2 sides and a contained angle are known (SAS) Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Steps to solve triangle Steps to solve triangle * If both the sine and cosine law can be used to solve a triangle, use the Sine Law, because it is easier !!! ** The side opposites to the largest angle in a triangle is the longest side and vice versa.   Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: A basketball backboard is supported by parallel brackets AB and CD, each 120 cm long. The brackets make an angle of 42° with the backboard. The distance from B to C is 100 cm. Find the length of the diagonal brace DB. Cosine Law 42° 120 cm Therefore the diagonal is approx. 81 cm long. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

a) Calculate CAB, correct to 1 decimal place. Example 2: Nick is flying a triangular course from A to B to C and back to A again. A is directly north of C. The distances he must travel are AB = 130 km, BC = 80 km, and CA = 76 km. a) Calculate CAB, correct to 1 decimal place. b) What bearing must he take from A to head toward B? Cosine Law A C N 130 km B 76 km W E S 80 km Therefore B is approx. S34.5 ° W of A. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Example 3 Label some angles A tunnel through the mountains is to be constructed to join A and B. Point C is 12.6 km from B. A cannot be seen from B or from C. Point D is 10.3 km from C and 6.7 km from A; ADC = 125° and DCB=142°. a) Find the length of the tunnel AB. Find the angle between i) AB and AD ii) AB and CB 6.7 km 12.6 km Label some angles 125° 142° 10.3 km Therefore the tunnel is approximately 24.2 km. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Example 3 A tunnel through the mountains is to be constructed to join A and B. Point C is 12.6 km from B. A cannot be seen from B or from C. Point D is 10.3 km from C and 6.7 km from A; ADC = 125° and DCB=142°. a) Find the length of the tunnel AB. Find the angle between i) AB and AD ii) AB and CB Angle between AB and AD. Angle between AB and CB. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

a) From the tower, what is the angle separating the aircraft? Example 4: The radar screen at an air traffic control tower shows a Piper Cherokee 15 km from the tower in a direction 30o east of north (N30oE), and a Cessna Skyhawk 16 km from the tower in a direction 40o east of north (N40oE), at their closest approach to each other. If the two aircraft are less than 2 km apart, the controller must file a report. a) From the tower, what is the angle separating the aircraft? b) Will the controller need to file a report? 15 km 30o a) The angle that separates the aircrafts is 10o. d 16 km 40o b) Will the controller need to file a report? 10° Let d be the distance between the aircrafts in km. The aircrafts are approximately 2.9 km apart and therefore does not need to file a report. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Example 5: Two cars A & B leave the same town. The first car A leaves at 12:30pm and drives 5o south of east at 100 km/h. The second car B leaves at 1:00pm and drives in a direction N40oW, at a speed of 120 km/h. How far apart are the cars at 3:00pm? Determine the bearing of Car B to Car A at 3:00pm. 40o B Let x be the distance between A & B at 3pm x At 100 km/h, 2.5 hours later (12:30pm – 3:00pm), 240 km A had travelled 250km (2.5 x 100km/h) Town At 120 km/h, 2 hours later (1:00pm – 3:00pm), 5o A B had travelled 240km (2 x 120km/h) s 85o Let’s label some angles 250 km Therefore the cars are approximately 453 km apart. Car B is N63oW of Car A. Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Day 1: Applications of Sine and Cosine Laws #1-7 Day 2: Applications of Sine and Cosine Laws #8 -13 Text: Check the website for updates Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

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