Year 2 Chapter 5: Radians Flashcards
. What is the difference between angles in degrees and angles in radians?
. Say how you convert degrees into rad and vice-versa:
. 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)
How many radians are in a circle (give reason)?
. 2pi radians - because the circumference of a circle = 2pi X R.
What does a sketch of ‘y= sin x’ look like (in radians)?
. 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 …….
What does a sketch of ‘y= cos x’ look like (in radians)?
. 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).
What does a sketch of ‘y= tan x’ look like (in radians)?
. 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 …….
What is sin(pi/6) equivalent to?
1/2
What is sin(pi/3) equivalent to?
Root(3) /2
What is sin(pi/4) equivalent to?
1/ root(2) = root(2) / 2
What is cos(pi/6) equivalent to?
Root(3) / 2
What is cos(pi/3) equivalent to?
1/2
What is cos(pi/4) equivalent to?
1/ root(2) = root(2) / 2
What is tan(pi/6) equivalent to?
1/ root(3) = root(3) / 3
What is tan(pi/3) equivalent to?
Root(3)
What is tan(pi/4) equivalent to?
1
What is the arc of a circle?
A part of the circumference of the circle.
Give the formula for the arc length of a circle (with notation):
. 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.
What is the sector of the circle (also say the difference between a minor and major sector)?
. 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.
Give the formula for the area of a sector of a circle (with notation):
. 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.
What is the segment of a circle?
. 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).
Give 2 formulas for the area of a segment of a circle (with notation):
. 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.
What are the three formulas for small angle approximations? And what do you need to remember about small angle approximations?
. 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.
