Year 2 Chapter 5: Radians

Question 1

. What is the difference between angles in degrees and angles in radians?
. Say how you convert degrees into rad and vice-versa:

Answer 1

. One degree represents 1/360 of a complete revolution or a circle.
. One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.
. y° = y rad X (pi/180)
. y rad = y° X (180/pi)

Question 2

How many radians are in a circle (give reason)?

Answer 2

. 2pi radians - because the circumference of a circle = 2pi X R.

Question 3

What does a sketch of ‘y= sin x’ look like (in radians)?

Answer 3

. Y/X intercept at origin.
. After origin, line curves up to Y=1, passes down through Y= 0, curves down to Y=-1, passes up through Y=0 then curves up to Y=1 again and continues this process.
. Y=1 at X=pi/2, X=5pi/2, X=9pi/2 ……
. Y=0 at X=0, X=pi, X=2pi ……. (so at any whole number multiplied by pi).
. Y=-1 at X=3pi/2, X=7pi/2, X=11pi/2 …….

Question 4

What does a sketch of ‘y= cos x’ look like (in radians)?

Answer 4

. Y-intercept at Y=1.
. After Y-intercept, line passes down through Y= 0, curves down to Y=-1, passes up through Y=0 then curves up to Y=1 again and continues this process.
. Y=1 at X=0, X=2pi, X=4pi ……. (so at any whole EVEN number multiplied by pi).
. Y=0 at X=pi/2, X=3pi/2, X=5pi/2 ……
. Y=-1 at X=pi, X=3pi, X=5pi ……. (so at any whole ODD number multiplied by pi).

Question 5

What does a sketch of ‘y= tan x’ look like (in radians)?

Answer 5

. Y intercept at origin.
. X intercept: a X pi (where ‘a’ is any whole number)
. Asymptote: x= a X pi/2 (where ‘a’ is any whole number).
. Line starts just after any asymptote at almost vertical gradient then gets less steep then goes through the next x-intercept at an almost flat gradient then gets steeper and ends at almost vertical gradient, tending towards next asymptote.
. All the lines have infinite ranges.
. Y=1 at X=pi/2, X=5pi/2, X=9pi/2 ……
. Y=0 at X=0, X=pi, X=2pi ……. (so at any whole number multiplied by pi).
. Y=-1 at X=3pi/2, X=7pi/2, X=11pi/2 …….

Question 6

What is sin(pi/6) equivalent to?

Answer 6

1/2

Question 7

What is sin(pi/3) equivalent to?

Answer 7

Root(3) /2

Question 8

What is sin(pi/4) equivalent to?

Answer 8

1/ root(2) = root(2) / 2

Question 9

What is cos(pi/6) equivalent to?

Answer 9

Root(3) / 2

Question 10

What is cos(pi/3) equivalent to?

Answer 10

1/2

Question 11

What is cos(pi/4) equivalent to?

Answer 11

1/ root(2) = root(2) / 2

Question 12

What is tan(pi/6) equivalent to?

Answer 12

1/ root(3) = root(3) / 3

Question 13

What is tan(pi/3) equivalent to?

Answer 13

Root(3)

Question 14

What is tan(pi/4) equivalent to?

Answer 14

1

Question 15

What is the arc of a circle?

Answer 15

A part of the circumference of the circle.

Question 16

Give the formula for the arc length of a circle (with notation):

Answer 16

. l=r(theta).
. l= arc length.
. r= radius of circle.
. theta= the angle, in radians, contained by the sector and joined to the centre of the circle.

Question 17

What is the sector of the circle (also say the difference between a minor and major sector)?

Answer 17

. The sector is an area of a circle enclosed by two radii and an arc (think pizza slice).
. The minor sector is the smallest sector out the two sectors and the major sector is the largest out of the two sectors.

Question 18

Give the formula for the area of a sector of a circle (with notation):

Answer 18

. A= 0.5(r²)(theta)
. r= radius of circle.
. theta= the angle, in radians, contained by the sector and joined to the centre of the circle.

Question 19

What is the segment of a circle?

Answer 19

. The segment is an area of a circle enclosed by a chord and an arc (looks like a semi-circle).
. It is also part of a sector of a circle but doesn’t have the triangle (which is made up of two radii and the same chord).

Question 20

Give 2 formulas for the area of a segment of a circle (with notation):

Answer 20

. A= 0.5(r²)(theta) - 0.5(r²)(sin(theta)).
. A= 0.5(r²) (theta - sin(theta))
. r= radius of circle.
. theta= the angle, in radians, contained by the sector and joined to the centre of the circle.

Question 21

What are the three formulas for small angle approximations? And what do you need to remember about small angle approximations?

Answer 21

. Sin(theta) = theta.
. Tan(theta) = theta.
. Cos(theta) = 1 - (theta)²/2.
. Only use the above 3 formulas when ‘theta’ is small (really close to 0) and measured in RADIANS.
. For Cos(x(theta)), the small angle approximation for it is ‘1- (x(theta))²/2’ instead of
‘1- x(theta)²/2’.
. If you’re told to find an approximate value of a small angle approximation, pretend ‘theta’ is 0.