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# Sine, cosine, radians

## 1.

С12 Sine, cosine rulesRadians

#sinerule #cosinerule #bearing #radian #arclength

#sectorarea #trianglearea

## 2.

Sine rule## 3.

ExerciseCalculate the value of and :

a)

b)

## 4.

ExerciseTown is 6 km, on a bearing of 020, from town . Town is

located on a bearing of 055 from town and on a bearing of

120 from town .

Work out the distance of town from

a) town

b) town

## 5.

Let us find :## 6.

ExerciseWork out the value of

a)

b)

## 7.

ExerciseWork out the value of and

a)

b)

## 8.

• When the angle to be found is larger than the given anglethere are two possible results, because you can draw two

different triangles which satisfy the task

• These results and are connected:

In this example:

(to 3 s.f.)

## 9.

ExerciseWork out the possible values of and the corresponding

values of .

## 10.

Cosine rule## 11.

ExerciseWork out the value of the third side:

## 12.

ExerciseFrom a point a boat sails due north for 7 km to . The boat leaves and

moves on a bearing of 100 for 10 km until it reaches .

Calculate the distance of from .

## 13.

ExerciseWork out the size of the angle marked with * (to 3 s.f.):

## 14.

ExerciseA helicopter flies on a bearing of 080 from to , where km. It

then flies for 60 km to a point .

Given that is 80 km from calculate the bearing of from .

## 15.

Data1 side, 2 angles

2 sides, angle

2 sides, angle

3 sides

To find

side

angle

side

angle

Formula

Sine rule

Sine rule

Cosine rule

Cosine rule

## 16.

Area## 17.

ExerciseWork out the area of the given triangle:

## 18.

ExerciseWork out the value of :

## 19.

Angle is equal to 1 radian, when the length of arc isequal to .

## 20.

radians ,so

radian

## 21.

The length of an arcwhen is in radians.

## 22.

ExerciseThe sector of the circle radius cm

contains an angle of radians. Find the

length of the arc, giving your answers in

the form , where and are integers.

## 23.

ExerciseReferring to the diagram, find:

a) The perimeter of the shaded region

when radians.

a) The value of when the perimeter of

the shaded region is 14 cm.

## 24.

ExerciseIn the diagram is the diameter of a circle, centre and radius 2 cm. The

point lies on the circumference such that radians.

a) State the value, in radians, of .

The shaded region is a template for a

brooch.

(b) Find the exact value of the

perimeter of the brooch.

## 25.

The area of a sectorwhere is in radians

## 26.

ExerciseWork out the area of the shaded region:

(a)

(b)

(c)

## 27.

ExerciseThe arc of a circle, centre and radius cm, is such that

radians.

Given that the perimeter of the minor sector is 30 cm:

(a) Calculate the value of .

(b) Show that the area of the minor sector is 36 .

(c) Calculate the area of the segment enclosed by the chord

and the minor arc .

## 28.

ExerciseThe diagram shows a triangular plot of land. The sides , and have

lengths 12, 14 and 10 m respectively.

The lawn is a sector of a circle, centre

and radius 6 m.

(a) Show that to 3 s.f.

(b) Calculate the area of the flowerbed.

## 29.

ExerciseIn the diagram, is the diameter of a circle, centre and

radius cm, and radians.

Given that the area of is equal to

that of the shaded segment,

show that