Angle TDA = Angle DTA = = 145 o = 10 o. Sin o o D. 35 o. 25 o 15 m

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1 T 10 o 36.5 The angle of elevation of the top of a building measured from point A is 25 o. At point D which is 15m closer to the building, the angle of elevation is 35 o Calculate the height of the building. Angle TDA = Angle DTA = TD Sin 15 o o 25 Sin10 15Sin25 TD Sin10 o B = 145 o = 10 o 36.5 m 35 o 145 o D Sin35 25 o 15 m o TB 36.5 A o TB 36.5Sin m

2 The angle of elevation of the top of a column measured from point A, is 20 o. The angle of elevation of the top of the statue is 25 o. Find the height of the statue when the measurements are taken 50 m from its base Angle BCA = o 50 Cos 20 AC 50 AC o Cos m (2 dp) = 70 o Angle ACT = = 110 o Angle ATC = = 65 o TC Sin o o 5 Sin Sin 5 TC 5.1 m(1 dp) o Sin65 65 o 110 o C 70 o T 5 o A 20 o 25 o 50 m B

3 A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (L). It then returns to harbour, a distance of 57 miles. (a) (b) Make a sketch of the journey. Find the bearing of the lighthouse from the harbour. (nearest degree) CosA x57x A 20.4 o Bearing o L 57 miles H A 40 miles 24 miles B

An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away.

4 An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles. Find the bearing of Q from point P. 670 miles Not to Scale P CosA b c a 2bc Q 530 miles CosP x530x670 P 48.7 o Bearing o 520 miles W

7 Complete Table Known Diagram Notes AAA AAS ASA SSA SAS SSS

8 The Law of COSINES

10 The Law of Cosines Use to find SIDES Use to find ANGLES a 2 = b 2 + c 2 2bcCosA cosa = b 2 + c 2 a 2 2bc

11 Use to find SIDES The Law of Cosines a 2 = b 2 + c 2 2bcCosA a = b 2 + c 2 2bcCosA

12 The Law of Cosines Use to find ANGLES cosa = b2 + c 2 a 2 2bc A = Cos 1 ( b2 +c 2 a 2 ) 2bc

13 Rewrite with different labels:

14 The Law of COSINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles: a 2 b 2 c 2 2bc cos A b 2 a 2 c 2 2accos B c 2 a 2 b 2 2abcosC

15 Use Law of COSINES when... you have 3 dimensions of a triangle and you need to find the other 3 dimensions. They cannot be just ANY 3 dimensions though, or you won t have enough information to solve the Law of Cosines equations. Use the Law of Cosines if you are given: SAS - 2 sides and their included angle SSS

16 While you wait: Solve the following triangles:

17 Law of Cosines Day 2

18 Example 1: Given SAS Solve triangle ABC, given that angle B = 98, side a = 13 and side c = 20. First draw a diagram. b 2 a 2 c 2 2accos B B 98 c = 20 b cos98 b C b A b 25.3

19 Example 1: Given SAS Now we have to find angles A and C. Let s take on angle A first. C a = 13 B 98 b = 25.3 c = 20 In order to find angle A should we use? a) Law of Sines b) Law of Cosines A c) Either law will work d) Neither will work

20 LOS vs LOC SSA might result in two possible solutions. But not in this case, since there is already an obtuse angle. If an angle might be obtuse, never use the Law of Sine equation to find it. Law of cosines is a better option.

21 Example 1: Given SAS Now that we know B and b, we can use the Law of Sines to find one of the missing angles: a = 13 B 98 c = 20 A = cos 1 ( ) C b = 25.3 A A~30. 6 Solution: b = 25.3, C = 51.5, A = 30.6

22 Example 1: Given SAS Now that we know B and b, we can use the Law of Sines to find one of the missing angles: C a = 13 B 98 Solution: b = 25.3 c = 20 b = 25.3, C = 51.5, A = 30.5 A 25.3 sin98 20 sin C 20sin 98 C sin C 51.5 A

23 Example 2: Given SAS Solve triangle, ABC, given that angle A = 39, side b = 20 and side c = 15. Use the Law of Cosines equation that uses b, c and A to find side a: B a 2 b 2 c 2 2bc cos A A c = b = 20 a C a cos39 a a 12.6

24 Example 2: Given SAS Use the Law of Sines to find one of the missing angles: A c = b = 20 B a = 12.6 C 12.6 sin39 15 sin C 15sin39 C sin C 48.5 B Important: Notice that we used the Law of Sine equation to find angle C rather than angle B. The Law of Sine equation will never produce an obtuse angle. If we had used the Law of Sine equation to find angle B we would have gotten 87.5, which is not correct, it is the reference angle for the correct answer, If an angle might be obtuse, never use the Law of Sine equation to find it.

25 Example 3: Given SSS Solve triangle, ABC, given that side a = 30, side b = 20 and side c = 15. A c = 15 b = 20 B a = 30 C We can use any of the Law of Cosine equations, filling in a, b & c and solving for one angle. Once we have an angle, we can either use another Law of Cosine equation to find another angle, or use the Law of Sines to find another angle.

26 Example 3: Given SSS Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it. For this reason, we will use the Law of Cosines to find the largest angle first (in case it happens to be obtuse). Angle A is largest because side a is largest: B A c = 15 b = 20 a = 30 C a 2 b 2 c 2 2bc cos A cos A cos A cos A cos A 275 A cos

27 Example 3: Given SSS B A c = 15 b = 20 a = 30 Solution: A = B = 36.3 C = 26.4 C Use Law of Sines to find angle B or C (its safe because they cannot be obtuse): 30 sin sin B 20sin117.3 B sin 1 30 B 36.3 C

28 The Law of Cosines a 2 b 2 c 2 2bc cos A b 2 a 2 c 2 2accos B c 2 a 2 b 2 2abcosC When given one of these dimension combinations, use the Law of Cosines to find one missing dimension and then use Law of Sines to find the rest. SAS SSS Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it.

32 Find all the angles created between each pair of cities. If the average speed is 49.8 mph, how long will the total trip take.

33 Repeat the process with 3 cities of your choice. 1) Choose 3 cities or locations. 2) Sketch a careful map of the three locations. Find the distance between each pair of cities include these values on your sketch. 3) Then find all the angles.

34 Homework 1) Complete the triangle table; 2) Three Cities! 3) Sec 9.4 page 352 #11, 13, all The above problems are suggested, do more if you need more practice.

35 Additional Resources Web Links:

Section 8.3 The Law of Cosines

147 Section 8.3 The Law of Cosines In this section, we will be solving SAS, SSS triangles. To help us do this, we will derive the Laws of Cosines. Objective 1: Derive the Laws of Cosines. To derive the