Angles

Angles:( denoted by Ð )

Angles are geometric figures made from two rays having the same endpoint The endpoint is called the vertex and the rays are called the sides of the angle. The side being rotated is called the initial side and the other side is the terminal side.

ÐABC, ÐCBA Ð B or q can be used to name angles.
The vertex is always in the middle of the angle.

Angles are measured in degrees or in radians. Degrees are the most common used unit for measuring angles, but in many modern applications the radians is used most often. Gradians is another unit of measure for angles which is commonly used in Europe and for surveying.

Angles are measured positive when they are rotated counter-clockwise and negative when rotated clockwise.

The initial side can be rotated several times before stopping at the terminal side.

 

 

Angles of Elevation and Depression

 

Degrees, Minutes, Seconds:

One degree, denoted by 1°, is 1/360 of a complete rotation counter-clockwise. ( 1 rotation = 360° )
One minute, denoted by 1', is 1/60 of a degree (1° = 60' ).
One second, denoted by 1", is 1/60 of a minute ( 1' = 60" ).

The degree measure of q = x° if q is rotated counter-clockwise and q = -x° if q is rotated clockwise.

 

 

Radians:

On the initial side of angle q, let P be a point one unit away from the vertex.
If q ³ 0, let s be the arc length traced by P as the initial side rotates through q; If q is negative , then the arc length s negative, then the radian measure of q is s.

q = s rad

If q is a complete rotation then q = 2p rad.

180° = p rad

 

 

Converting Degrees to Radians:

The conversion of degrees to radians is just a proportion problem.
ex.
Convert 60° to radians.

60/360 = X/2p
2p (60) = 360x
2(3.14)(60) = 360x
1.0467 = x
60° = 1.0467 rads

If P is a point on the initial side of angle q, let P be r units from the vertex, as the initial side of angle q is rotated to the terminal side, and P traces an arc length s then:

 

From now on, q = s will be understood to be in radians, except in places where "rad" is needed for clarity.

 

Trigonometric Functions:

The sum of all three angles of any triangle is 180° = p rad

Pythagorean Theorem:

c²= a²+ b²

sine, cosine, tangent.

Secant: 

sec q = 1/cos q

Cosecant:

csc q = 1/sin q

Cotangent:

cot q = 1/ tan q

cot q = cos q/sin q

tan q = sin q/cos q

and

sin² q + cos²q = 1

 

 

Trigonometric Functions Using Coordinate Systems:

Angle q is in standard position if its vertex is at the origin and its initial side is on the x-axis.

"q is in Q1 " means that angle q is in standard position and its terminal side is in quadrant 1.

If the terminal side is on an axis not in a quadrant, this angle is called a quadrantal angle or a between quadrant angle.

If q is in Q1 and it is an acute angle, then the previous trigonometric functions can be applied.

Let P = (x , y) be any point on the terminal side of q , and the distance r from the origin to point P is:

 

 

Reference Angles:

The reference angle of q is smallest angle j between the terminal side of q and the x-axis.

Now, let P be a point on the terminal side of the angle and drop a perpendicular from P to the x-axis. By doing this, a natural right triangle is formed.

j is the angle between the leg along the x-axis and the hypotenuse and is also the reference angle of q.

If P has coordinates (x,y) and then the right triangle has dimensions:

The trig. functions j can be compared to the trig. functions of q, which are:

If j is the reference angle of an angle q, then the value of a trig. function of q is agreeable with the value of the same trig. function of j, except possibly for the sign.

 

 

Graph of Trig. Functions:

Consider x as a real number or an angle in radians.
If f is any trig. function then f( x + 2p ) = f(x) ; Therefore the graph of the function has an interval of 2p.

Graph of f(x) = sin x :

Graph of f(x) = cos x :

Graph of f(x) = csc x :

Graph of f(x) = sec x :

The functions above are periodic of period 2p.

Graph of f(x) = tan x :

Graph of f(x) = cot x :

The functions above are periodic of period p.

 

 

 

The graph of the functions : (waves)

f(x) = a sin ( bx + c ) and f(x) = a cos ( bx + c )

| a | is the amplitude

2p/b is the period

c/b is the phase shift

 

 

Simple Harmonic Motion:

An event is said to be a simple harmonic motion if it can be described by:

y = a sin ( bx - c )

where b > 0 and x is the distance. If so, the amplitude is | a | , the wavelength, denoted by l (the Greek letter lambda), l = 2p/b and the phase shift is c / b.

Frequency = 1/period

Frequency ´ period = speed

 

 

 

Inverse Trigonometric Functions and their Graphs:

A function has an inverse function strictly when no horizontal line intersects the graph more than once. Since trig. functions don't have inverse functions, and it is useful to have an inverse function, a restriction is applied to the domain so that an inverse function might exist. Certain conditions must be met in order for the trig. functions to have an inverse function.

   1. Each value of the range is only taken on once so it can pass the horizontal line test.
   2. The range of the function with its restricted domain is the same as the range of the original function.
   3. The domain includes the most commonly used numbers (or angles)
      0 < x < p/2.
   4. The graph is connected (if possible).

 
Inverse sine:

If -1 < x < 1, then f(x) = sin^-1 x ( or f(x) = arcsin x), if and only if sin f(x) = x and - p/2 < f(x) < p/2.

Inverse cosine:

If -1 < x < 1, then f(x) = cos^-1x ( or f(x) = arccos x), if and only if cos f(x) = x and
0 < f(x) < p.

Inverse tangent:

If x is any real number, then f(x) = tan^-1 x ( or f(x) = arctan x), if and
only if tan f(x) = x and - p/2 < f(x) < p/2.

Inverse cosecant:

If | x | ³ 1, then f(x) = csc^-1 x ( or f(x) = arccsc x), if and only if
csc f(x) = x and - p/2 < f(x) < p/2 , y ¹ 0.

Inverse secant:

If | x | ³ 1, then f(x) = sec^-1 x ( or f(x) = arcsec x), if and only if
sec f(x) = x and 0 < f(x) < p, y ¹ p/2.

Inverse cotangent:

If x is any real number, then f(x) = cot^-1 x ( or f(x) = arccot x), if and only if cot f(x) = x and 0 < f(x)< p.

 

The Addition Formulas:

sin ( a + b ) = sin a cos b + cos a sin b

sin ( a - b ) = sin a cos b - cos a sin b

cos ( a + b ) = cos a cos b - cos a cos b

cos ( a - b ) = cos a cos b + cos a cos b

 

 

Double-Angle Formulas:

sin 2a = 2 sin a cos a

cos 2a = cos² a - sin² a
= 1 - 2sin² a
= 2cos² a - 1

tan 2a = (2tan a) / (1-tan² a)

 

 

Half-Angle Formulas:

 

The Area Of A Triangle:

Given any triangle:

If two sides, A and C, are given and the included angle b then the area of the triangle is :

Area= ½ AC sin b

 

If all three angles and a side of a triangle are given, then the area of the triangle is:

If all three sides are known and the semiperimeters,
[ s = ½ (A + B + C)] then the area is :

 

 

The Law of Cosines:

Given a triangle:

The law of cosines are:

A² = B² + C² - 2 BC cos a

B² = A² + C² - 2 AC cos b

C² = A² + B² - 2 AB cos c

 

 

The Law of Sines:

In a triangle with sides A, B and C and opposite angles a, b and c ,respectively,

 

 

From the Trig. Identities:

Rectangular coordinate can be converted to polar coordinates and vice versa.

Polar to Rectangular: ( r , q ) to ( x , y )

x = r cos q
y = r sin q if r ³ 0

Rectangular to Polar: ( x , y ) to ( r , q )

 

 

Polar Coordinates

The origin or pole of polar coordinates called point O and a horizontal half line with the endpoint O, called the polar axis, is a nonnegative line which the length of a segment can be measured. Angle q is the angle in which the terminal side is rotated.

Point P has polar coordinates ( r,q ), r ³ 0, if q has the polar axis as the initial side and point P is on the terminal side of q and |OP| = r, the distance from the origin to P is r.

Point P has polar coordinates ( r,q ), r < 0 if the terminal side of q is extended | r | units to the other side of the origin to point P. ( when r is negative)

ex.

 

 

Head to Tail Rule:

Given two unequal vectors:

1. Place the initial point of one vector at endpoint of the other vector.

2. Make a triangle from the joined vectors and the third vector is the vector sum of the two vectors.

If two vectors having the same direction but different magnitude, their vector sum is the sum of the magnitudes in the same direction.

If two vectors have the same magnitude and are in opposite directions their vector sum is a zero vector.

If two vectors are in opposite direction and have different magnitudes, then their vector sum is the difference in magnitudes of the vectors in the direction of the vector with the larger magnitude.

 

The Parallelogram Rule:

Given two vectors, c and d, not equal to each other:

1. Place the initial points of both vectors in the same point and call it point O.

2. Draw a line parallel to vector c at the endpoint of vector d and also draw a line parallel to vector d at the endpoint of vector c, name the point of intersection of the parallel lines, call it point B.

3. `OB is the vector sum of vector c and vector d.

 

Vectors

A vector is a quantity that has both direction and magnitude. A vector is denoted by a directed line segment. The magnitude of a vector is the length of the segment and the direction of a vector is where the segment is pointing. Vectors are named like segments or by small letters.

ex.

Point A is the initial point and point B is the terminal point of the vector above.

A vector that has no magnitude but has a direction is called a zero vector.

Vectors are equal when their magnitudes and direction are equal.

 

 

 

Trig. Form Of Complex Numbers:

In the complex number a + bi, where a is the real part and b is the imaginary part. ( both a and b are real numbers) The complex number can be graphed in the complex plane, which is similar to the rectangular plane, the real part are the points along the x-axis and the imaginary part are the points along the y-axis.

Complex numbers are the ordered pair (a , b).

With this, complex numbers can be treated in polar and rectangular coordinates.

Complex numbers can also be converted to vectors. Just let the complex number equal a vector variable, and the complex number can be manipulated in many different ways.

ex.
z = a + b i

From trig. identities:
a = r cos q
b = r sin q

z = r ( cos q + i sin q )

NOTE:
cos q + i sin q is abbreviated as cis q .

Thus, z = r cis q .

Addition of complex numbers using vectors:

ex.
s = 2 + 3i
t = -5 + 2i
s + t = (2 + 3i) + ( -5 + 2i) = -3 + 5i

Multiplication of complex numbers in trig. form:

(r cis q)(s cis j) = rs cis ( q + j )

Reciprocal of complex numbers in trig. form:

(r cis q)^-1= r^-1 cis (-q)

for any integer n then;

for k = 0,1,2,3,4, . . . , n - 1

 

Hyperbolic Functions

Hyperbolic Functions are defined in terms of exponential functions but are like trig. functions in many ways.

 

Inverse Hyperbolic Functions

Since hyperbolic functions are defined in terms of exponential functions, it is expected that their inverses can be expressed in the inverse of their exponential functions:

Inverse Hyperbolic Functions in terms of logarithms: