Chapter 9 Nonright Angled Triangle Trigonometry Opening problem

Chapter 9 Non-right Angled Triangle Trigonometry

Opening problem A triangular sail is to be cut from a section of cloth. Two of the sides must have lengths 4 m and 6 m as illustrated. The total area for the sail must be 11. 6 m 2, the maximum allowed for the boat to race in its class. a. Can you find the size of the angle q between the two sides of given length? b. Can you find the length of the third side of the sail?

6 q 4

Included angle Knowing two sides and the included angle between them 8 cm 49 o 10 cm

No angles Knowing all three sides 9 cm 8 cm 10 cm

Using the included angle A c B b h 49 o C 10 cm a

A c h b 180 o - C B a C D

Exercise 9 A #1

Exercise 9 A #3 6. A parallelogram has two adjacent sides of length 4 cm and 6 cm respectively. If the included angle measures 52 o , find the area of the 6 cm parallelogram. 4 cm 52 o h

A.

B.

C.

Cosine rule The cosine rule involves the sides and angles of a triangle. A b c B a C

The cosine rule can be used to solve problems involving triangles given: • two sides and an included angle • three sides.

Ambiguous case If we are given two sides and a non-included angle, then we try to find the third side we obtain a quadratic equation. This is an ambiguous case where there may be two plausible solutions. We may not be able to solve for the length uniquely if there are two positive, plausible solutions to the quadratic equation. b a b B a B

Exercise 9 B #1 a A 105 o 21 cm C 15 cm B

Exercise 9 B #4 11. a. Find the smallest angle of a triangle with sides 11 cm, 13 cm and 17 cm. 11 cm 13 cm q 17 cm

b. Find the largest angle of a triangle with sides 4 cm, 7 cm and 9 cm. 4 cm q 9 cm 7 cm

Exercise 9 B#5 a. Find cos q but not q. 5 cm 2 cm q 4 cm b. Find the value of x x cm 1 cm

Exercise 9 B #6

The Sine Rule The sine rule is a set of equations which connects the lengths of the sides of any triangle with the sines of the angles of the triangle. The triangle does not have to be right angled for the sine rule to be used. b A C c a B

Exercise 9 C. 1#2 15. In triangle ABC find: a. a if A = 63 o , B = 49 o and b = 18 cm

b. b if A = 82 o , C = 25 o and c = 34 cm A + B + C = 180 o 82 o + B + 25 o = 180 B = 73 o

c. c if B = 21 o , C = 48 o and a = 6. 4 cm. A + B + C = 180 o A + 21 o + 48 o = 180 A = 111 o

Ambiguous case The problem of finding angles using the sine rule is more complicated because there may be two possible answers. For example, We call this situation an ambiguous case. This occurs when we have two sides and a non-included angle. investigation: the ambiguous case.

Exercise 9 C. 2#1 17. Triangle ABC has angle B = 40 o, b = 8 cm, and c = 11 cm. Find the two possible values for angle C.

11 11 B 8 8 8 B 40 o ACUTE 40 o C C

ym 95 o xm 30 o 118 o 22 m

Use the cosine rule when given: • three sides • two sides and an included angle. Use the sine rule when given: • one side and two angles • two sides and a non-included angle, but beware of an ambiguous case which can occur when the smaller of the two given sides is opposite the given angle.

20. Rodrigo wishes to determine the height of a flagpole. He takes a sighting to the top of the flagpole from point P. He then moves further away from the flagpole by 20 meters to point Q and takes a second sighting. The information is shown in the diagram alongside. How high is the flagpole?

Q 28 o 20 m P 53 o

21. A golfer played his tee shot a distance of 220 m to point A. He then played a 165 m six iron to the green. If the distance from tee to green is 340 m, determine the number of degrees the golfer was off line with his tee shot.