# MST124 BOOK B UNIT 4

- Created by: Scar.Rose.1
- Created on: 05-06-22 10:12

## 1

**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

## 2

**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

## 3

inverse sine = arcsine = arcsin(x) = sin^{-1}x

inverse cosine = arccosine = arccos(x) = cos^{-1}x

inverse tangent = arctangent = arctan = tan^{-1}x^{ }

**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)

sin^{2}(angle) + cos^{2}(angle) = 1

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

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

## 4

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)

## 5

**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

## 6

**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^{-1}x = y (where y is the number in the interval [-pi/2, pi/2] such that siny = x

## 7

**Inverse cosine**cos

^{-1 }with the domain [-1,1] and rule: cos

^{-1}x = 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

^{-1}x = y (where y is the number in the interval (-pi/2, pi/2) such that tan y = x

sin^{-1}(-x) = -sin^{-1}x

cos^{-1}(-x) = pi - cos^{-1}x

tan^{-1}(-x) = -tan^{-1}x

- 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

## 8

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

## 9

**Cosine rule**

a^{2} = b^{2} + c^{2} - (2bc*cosA)

b^{2} = c^{2} + a^{2} - (2ca*cosB)

c^{2} = a^{2} + b^{2} - (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)

## 10

**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)

sin^{2}(theta) + cos^{2}(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)

## 11

(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

tan^{2}(theta) + 1 = sec^{2}(theta)

1 + cot^{2}(theta) = cosec^{2}(theta)

**Angle sum identities for sine and cosine**sin(A + B) = sinAcosB + cosAsinB

cos(A + B) = cosAcosB - sinAsinB

## 12

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) = cos

^{2}(theta) - sin

^{2}(theta)

tan(2*theta) = (2*tan(theta))/1-tan

^{2}(theta)

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

^{2}(theta)

cos(2*theta) = 2*cos

^{2}(theta) - 1

## 13

**Half-angle identities **sin

^{2}(theta) = 1/2(1 - cos(2*theta))

cos

^{2}(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)

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