Everything Maths Grade 11

CHAPTER 17. TRIGONOMETRY 17.5

Exercise 17 - 11

1. Show that
   sinAˆ
   a

=

sinBˆ

b

=

sinCˆ

c

is equivalent to:

a

sinAˆ

=

b

sinBˆ

=

c

sinCˆ

Note: either of these two forms can be used.

2. Find all the unknownsides and angles of the following triangles:

(a)�P QR in whichQˆ = 64◦;Rˆ = 24◦and r = 3

(b)�KLM in whichKˆ = 43◦;Mˆ = 50◦and m = 1

(c)�ABC in whichAˆ = 32, 7 ◦;Cˆ = 70, 5 ◦and a = 52, 3

(d)�XY Z in whichXˆ = 56◦;Zˆ = 40◦and x = 50

3. In�ABC,Aˆ = 116◦;Cˆ = 32◦and AC = 23 m. Find the length of the side AB.


4. In�RST,Rˆ = 19◦;Sˆ = 30◦and RT = 120 km. Find the length of the side ST.


5. In�KMS,Kˆ = 20◦;Mˆ = 100◦and s = 23 cm. Find the length of the side m.

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(1.) 014v (2.) 014w (3.) 014x (4.) 014y (5.) 014z

The Cosine Rule EMBDO

DEFINITION: The Cosine Rule

The cosine rule appliesto any triangle and states that:

a^2 = b^2 + c^2 − 2 bccosAˆ

b^2 = c^2 + a^2 − 2 cacosBˆ

c^2 = a^2 + b^2 − 2 abcosCˆ

where a is the side oppositeAˆ, b is the side oppositeBˆ and c is the side oppositeCˆ.

The cosine rule relates the length of a side of atriangle to the angle opposite it and the lengths of the
other two sides.

Consider�ABC which we will use to show that:

a^2 = b^2 + c^2 − 2 bccosA.ˆ