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Middle Years

12.01 Sine rule

Worksheet
Sine rule
1

For each of the given triangles, determine if there is enough information to find all the remaining sides and angles in the triangle using only the sine rule:

a

Three sides are known:

b

Two of the angles and the side included between them are known:

c

Two of the angles and a side not included between them are known:

d

Two of the sides and an angle included between them are known:

2

For each of the following triangles, write an equation relating the sides and angles using the sine rule:

a
b
3

Consider the following diagram:

a

Find an expression for \sin A in \triangle ACD.

b

Find an expression for \sin B in \triangle BDC. Then make x the subject of the equation.

c

Substitute your expression for x into the equation from part (a), and rearrange the equation to form the sine rule.

Unknown sides
4

For each of the following triangles, find the side length a using the sine rule. Round your answers to two decimal places.

a
b
c
d
e
5

For each of the following triangles, find the length of side x, correct to one decimal place:

a
b
6

For each of the following triangles:

i

Find the value of a using the sine rule. Round your answer to two decimal places.

ii

Use another trigonometric ratio and the fact that the triangle is right-angled to calculate and confirm the value of a. Round your answer to two decimal places.

a
b
c
7

Consider the following triangle:

a

Find the length of side HK to two decimal places.

b

Find the length of side KJ to two decimal places.

8

Consider the triangle with \angle C = 72.53 \degree and \angle B = 31.69 \degree, and one side length a = 5.816\text{ m}.

a

Find \angle A. Round your answer to two decimal places.

b

Find the length of side b. Round your answer to three decimal places.

c

Find the length of side c. Round your answer to three decimal places.

Unknown angles
9

For each of the following diagrams, find the value of the angle x using the sine rule. Round your answers to one decimal place.

a
b
c
d
e
f
g
h
i
j
10

For each of the following acute angled triangles, calculate the size of\angle B to the nearest degree:

a

\triangle ABC where \angle A = 57 \degree side a = 156 \text{ cm} and side b = 179 \text{ cm}

b

\triangle ABC where \angle A = 48 \degree side a = 2.7 \text{ cm} and side b = 1.9 \text{ cm}

11

The angle of depression from J to M is 68 \degree. The length of JK is 25 \text{ m} and the length of MK is 28 \text{ m} as shown:

Find the following, rounding your answers to two decimal places:

a

Find x, the size of \angle JMK.

b

Find the angle of elevation from M to K.

12

Consider \triangle ABC below:

a

Find x to the nearest degree.

b

Find \angle ADB to the nearest degree.

13

Consider the following diagram of a quadrilateral:

Find the value of \theta, correct to two decimal places.

Applications
14

Consider the following diagram:

a

Find the size of \angle OBA.

b

Find the length of k, to two decimal places.

15

A bridge connects two towns on either side of a gorge, where one side of the gorge is inclined at 59 \degree and the other side is inclined at 70 \degree. The length of the steeper incline is 59.1 \text{ m}.

Find x, the length of the bridge. Round your answer correct to one decimal place.

16

During football training, the coach marks out the perimeter of a triangular course that players need to run around. The diagram shows some measurements taken of the course, where side length a = 14 \text{ m}:

a

Find the size of \angle A.

b

Find the length of side c, correct to two decimal places.

c

Find the length of side b, correct to two decimal places.

d

Each player must sprint one lap and then jog one lap around the triangle. This process is to be repeated 3 times by each player.

If Tara can run 280 \text{ m/min}, and can jog at half the speed she runs, calculate the time this exercise will take her, correct to one decimal place.

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