MST124 BOOK B UNIT 4

Radians
One radian is the angle subtended at the center of a circle by an arc that has the same length as the radius 
360^o = 2*pi radians 
1 radian = 360/2*pi = 180/pi = ~57^o  
0^o = 0 radians 
30^o = pi/6
45^o = pi/4
60^o = pi/3
90^o = pi/2
180^o = pi
360^o = 2pi 

Converting between degrees and radians 
number of radians = (pi/180) x number of degrees
number of degrees = (180/pi) x number of radians

Length of an arc of a circle
arc length = r*theta
where r is the radius of the circle and theta is the angle subtended by the arc, measured in radians 

Area of a sector of a circle
area of sector = 1/2*r*theta
- where r is the radius of the circle and theta is the angle of the sector, measured in radians 

Pythagoras' theorem
For right-angled triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides 

Sine, cosine and tangent
sine of angle = sin(angle) = opposite/hypotenuse
cosine of angle = cos(angle) = adjacent/hypotenuse
tangent of angle = tan(angle) = opposite/adjacent 

inverse sine = arcsine = arcsin(x) = sin^-1x 
inverse cosine = arccosine = arccos(x) = cos^-1x
inverse tangent = arctangent = arctan = tan^-1x 

Angle in radians - angle in degrees - sin(angle) - cos(angle) - tan(angle)
pi/6 - 30 - 1/2 - (sqrt3)/2 - 1/(sqrt3)
pi/4 - 45 - 1/(sqrt2) - 1/(sqrt2) - 1
pi/3 - 60 - (sqrt3)/2 - 1/2 - sqrt3 

Trigonometric identities
tan(angle) = sin(angle)/cos(angle) 
sin²(angle) + cos²(angle) = 1
sin((pi/2) - theta) = cos(theta)
cos((pi/2 - theta) = sin(theta) 

Sine, cosine and tangent
Suppose that theta is any angle and (x,y) are the coordinates of its associated point P on the unit circle.
sin(theta) = y and cos(theta) = x
provided that x doesn't equal 0,
tan(theta) = y/x
if x = 0 then tan(theta) is undefined 

First quadrant - A for all +ve
Second quadrant - S for sin +ve
Third quadrant - T for tan +ve
Fourth quadrant - C for cos +ve

Suppose that theta is an angle whose associated point P does not lie on either the x- or y-axis, and phi is the acute angle between OP and the x-axis. Then, sin(theta) = ±sin(phi), cos(theta) = ±cos(phi) and tan(theta) = ±tan(phi)

Sine Graph
- as the angle increases from 0 to 2*pi, the value of sin(theta) oscillates from 0 to 1 to 0 to -1 and back to 0 
- this is because sin(theta) is the y-coordinate of the point P on the unit circle
- as the angle increases from 0 to 2*pi, the point P starts on the positive x-axis and rotates anticlockwise around the unit circle, back the where it started 
- as P rotates, its y-coordinate oscillates from 0 to 1 to 0 to -1 to 0
- sin(theta) is never greater than 1 or less than -1 
- since P returns to its original position after a complete rotation through 2*pi, sin(theta+2*pi) = sin(theta) for any angle
- the graph oscillates endlessly to the right and to the left, repeating its shape after every interval of 2*pi
- the graph is periodic with a period 2*pi

Cosine Graph
- cos(theta) is the x-coordinate of the point P on the unit circle 
- like the sine graph, the graph of cosine is periodic with period 2*pi
- any vertical line through the center of a peak or trough is a line of mirror symmetry
- you can obtain the graph of cosine by translating the graph of sine to the left by pi/2 

Tangent Graph
- broken up into separate pieces
- takes arbitrarily large +ve values and arbitrarily large -ve values 
- periodic with a period of pi rather than 2*pi
- the breaks in the graph occur when theta = pi/2 + n*pi for some integer n
- values for theta for which tan(theta) is undefined, because y = 0 
- graph also has asymptotes 

- sine, cosine, and tangent functions are not one-to-one
Inverse sine function
sin^-1, with the domain [-1,1] and rule: sin^-1x = y (where y is the number in the interval [-pi/2, pi/2] such that siny = x

Inverse cosine
cos^-1 with the domain [-1,1] and rule: cos^-1x = y (where y is the number in the interval [0, pi] such that cosy = x) 

Inverse tangent
tan^-1 with the domain R and rule: tan^-1x = y (where y is the number in the interval (-pi/2, pi/2) such that tan y = x

sin^-1(-x) = -sin^-1x
cos^-1(-x) = pi - cos^-1x
tan^-1(-x) = -tan^-1x 

- generally, when you want to solve a simple trig equation, its best to begin by finding all the solutions that lie in an interval of length 2*pi sich as [0, 2*pi] or [-pi, pi] 
- there are usually two such solutions, and then you can obtain other solutions that you want by adding integer multiples of 2*pi to the first solutions you found

Lower case letters = sides of triangle, upper case letters = angles 
Sin rule
a/sinA = b/sinB = c/sinC 

Using the sine rule to find a side length
You can use the sine rule to find an unknown side length of a triangle if you know:
- the opposite angle
- another side length and its opposite angle 

- using the sine rule, there are two possible values for the angle, sinA and 180 (or pi) - sinA 
- you would get an acute and obtuse angle
- you need more info to pick the right one 

Using the sine rule to find an angle
You can use the sine rule to find an unknown angle in a triangle if you know:
- the opposite side length
- another side length and its opposite angle
- sometimes you also need to known if the unknown angle is obtuse or acute 

Cosine rule
a² = b² + c² - (2bc*cosA) 
b² = c² + a² - (2ca*cosB)
c² = a² + b² - (2ab*cosC)

Using the cosine rule to find a side length
You can use the cosine rule to find an unknown side length of a triangle if you know the other two side lengths and the angle between them 

Using the cosine rule to find an angle
You can use the cosine rule to find an unknown angle if you know all three side lengths 

Area of a triangle 
area = 1/2 x base x height 
or
For a triangle with an angle theta between two sides of lengths a and b:
area = 1/2*a*b*sin(theta) 

Gradient and angle of inclination of a straight line
For any non-vertical straight line with angle of inclination theta,
gradient = tan(theta)
The angle of inclination is measured when the line is drawn on axes with equal scales 

tan(theta) = sin(theta)/cos(theta) 
sin²(theta) + cos²(theta) = 1

sin(theta + 2*pi) = sin(theta)
cos(theta + 2*pi) = cos(theta)
tan(theta + pi) = tan(theta) 

sin(-theta) = -sin(theta)
cos(-theta) = cos(theta)
tan(-theta) = -tan(theta) 

sin(pi/2 - theta) = cos(theta)
cos(pi/2 - theta) = sin(theta) 

(cosecant) cosec(theta) = hypotenuse/opposite = 1/sin(theta)
(secant) sec(theta) = hypotenuse/adjacent = 1/cos(theta)
(cotangent) cot(theta) = adjacent/hypotenuse = cos(theta)/sin(theta) 

tan(theta) = sin(theta)/cos(theta)
so
cot(theta) = 1/tan(theta) 

- cosecant, secant and cotangent functions are reciprocals of the sine, cosinse and tangent functions

tan²(theta) + 1 = sec²(theta)
1 + cot²(theta) = cosec²(theta)

Angle sum identities for sine and cosine
sin(A + B) = sinAcosB + cosAsinB
cos(A + B) = cosAcosB - sinAsinB 

tan(A + B) = (sinAcosB + cosAsinB)/(cosAcosB - sinAsinB)
= (tanA + tanB)/1-(tanAtanB) 

Angle difference identities
sin(A - B) = sinAcosB - cosAsinB
cos(A - B) = cosAcosB + sinAsinB
tan(A - B) = (tanA - tanB)/1+tanAtanB 

Double-angle identities
sin(2*theta) = 2sin(theta)*cos(theta)
cos(2*theta) = cos²(theta) - sin²(theta)
tan(2*theta) = (2*tan(theta))/1-tan²(theta) 

Alternative double-angle identities for cosine
cos(2*theta) = 1 - 2*sin²(theta)
cos(2*theta) = 2*cos²(theta) - 1

Half-angle identities 
sin²(theta) = 1/2(1 - cos(2*theta))
cos²(theta) = 1/2(1 + cos(2*theta))

sinA + sinB = 2sin(A+B/2)*cos(A-B/2)
sinA - sinB = 2sin(A-B/2)*cos(A+B/2) 
cosA + cosB = 2cos(A+B/2)*cos(A-B/2)
cosA - cosB = -2sin(A+B/2)*sin(A-B/2)