Radian measure Objective Deadlines / Progress

Rad

ian

Mea

sure

Convert degrees to radians and back

Find arc lengths, sector and segment areas using radian measure; solve problems working back and forward using formulae for arc length and sector area;Use the sine and cosine rules with radians – usually with arc and sector problems Sketch Trigonometric graphs using radians; sketch transformations of Trigonometric graphs Solving Trigonometric equations finding all solutions in a given range in radian measure

Rec

ipro

cal a

nd in

vers

e fu

nctio

ns Graphs and transformations of reciprocal

Trigonometric functions Trigonometric Identities with reciprocal functions;Basic proofs and equations with reciprocal functionsGraphs of inverse Trigonometric functions and transformationsSolve harder Trig equations

Add

ition

form

ulas

Introduction to addition formulas Double angle formulas Half angle formulasFormulas rearranged for sin2 x and cos2 xSolving equations Proofsexpressions for acos θ + bsin θ in the equivalent forms of Rcos(θ ±α) or Rsin(θ ±α); including max and min values

Use trigonometric functions to solve problems in context, including models involving vectors, kinematics and forces

standard small angle approximations of sine, cosine and tangentGradient of sin x and cos x from first principles

1

Radian measure

Radians Degreesπc=180 The ❑c is often unwritten

For example, 60 °=60× π180

=π3radians

For example the range 180 °≤ x≤360 ° is equivalent to π ≤x ≤2π

Formulas: either learn these (really learn them) or do each problem starting from circumference or area of the relevant circle

Area sector = 12θ r2

Arc length = rθ

Segment area = 12r2 (θ−sin θ )

Reminder: Sine rule a

sin A= b

sinB= c

sinC Area rule 12ab sinC

Cosine rule a2=b2+c2−2bc cosA

2

Radian measure

WB1

Convert to degrees

a) 1.8c b) 7π8

c

c) 4 π15

c

Convert to

d) 150 ° e) 110° f) 35 °

WB2 Arc AB of a circle, with centre O and radius r, subtends an angle of θ radians at O. The

Perimeter of sector AOB is 10 cm. Express r in terms of θ.

3

r

r

θ

A

B

O rθ

Radian measure

WB3Arc length

The border of a garden pond consists of a straight edge AB of length 2.4m, and a curved part C, as shown in the diagram below. The curved part is an arc of a circle, centre O and radius 2m. Find the length of C.

WB4Sector area

In the diagram, the area of the minor sector AOB is 28.9cm2. Given that angle AOB is 0.8 rad, calculate the value of r.

4

O

A B2.4m

2m

C

θ

B

O 0.8c

Ar cm

Radian measure

WB5Sector area

A plot of land is in the shape of a sector of a circle of radius 55m. The length of fencing that is needed to enclose the land is 176m. Calculate the area of the plot of land.

WB6 Segment area

Calculate the Area of the segment shown in the diagram below

5

O

π 3

2.5cm

Radian measure

WB7 Segment area

In the diagram AB is the diameter of a circle of radius r cm, and angle BOC = θ radians. Given that the Area of triangle AOC is 3 times that of the shaded segment, show that 3θ – 4sinθ = 0

WB8 Figure 2 shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 63 m,(a) find the exact size of angle PQR in radians.(b) Show that the area of the patio PQRS is 12 m2.(c) Find the exact area of the triangle PQR.(d) Find, in m2 to 1 decimal place, the area of the segment PRS.(e) Find, in m to 1 decimal place, the perimeter of the patio PQRS.

6

Radian measure

WB9 The diagram shows a triangle ABC and the arc AB of a circle whose centre is C and radius is 24 cm. the length of AB is 32 cm. The size of angle ACB is θ °

a) Show that θ=1.46c correct to 3 sfb) Calculate the length of arc AB to the nearest cmc)

i. Calculate the area of the sector ABC to the nearest cm2

ii. Calculate the area of the shaded segment to the nearest cm2

WB10 The diagram shows a triangle ABC, where angle BAC is 0.4 radians. BAD is a sector of the circle with centre A and radius AB

a) The area of sector BAD is to the nearest 5cm2 show that length AB is 5 cmb) The area of triangle ABC s twice the area of sector BAD, find length ACc) Find the perimeter of region BCD

7