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Transcript

Toka Emad El-din

Ms.Basma Mohamed El-Sayed

INTRODUCTION

Introduction

Trigonometric Function

Trigonometric functions are elementary functions, the argument of which is an angle. Trigonometric functions describe the relation between the sides and angles of a right triangle. Applications of trigonometric functions are extremely diverse.

The trigonometric functions can be defined using the unit circle. The figure shows a circle of radius r=1. There is a point M(x,y) on the circle. The angle between the radius vector OM and the positive direction of the x-axis is equal to .

Graph

of Trigonometric Function

A function of an angle, or of an abstract quantity, used in trigonometry, including the sine, cosine, tangent, cotangent, secant, and cosecant, and their hyperbolic counterparts.

REAL-LIFE APPLICATIONS

REAL-LIFE

APPLICATIONS

Sound

Sine waves, when extended across the horizontal axis, look much like the pattern of a regular sound wave. The wave can be manipulated by adding things to the sine function, making the wave taller, wider, or narrower. Sound engineers use this mathematical manipulation of sine waves to create different computer-generated sounds. More complicated trigonometry is involved to create rich, realistic-sounding music.

Sound

Wave

Ocean Wave

Ocean

Wave

Tide change can be found in trigonometry by using the height of the water from a given locations during its high and low tides. You also need the height of the water at both high and low tides. Oceanographers commonly use a sinusoidal graph to display high and low tides.

Ocean waves are very important for weather forecasting and climate modeling as well as for coastal communities, shipping routes and offshore industry. Recent studies of coupling atmosphere-ocean-wave models have shown improvements in the simulation of North Atlantic sea surface temperatures in climate models.

Seismology

Seismology graphs uses sine and cosine to calculate the magnitude of an earthquake. This measures the severity of the earthquake. Trigonometry is used to calculate the vertical and horizontal distances traveled by the seismic waves .This helps the seismologist to record how strong were previous earthquakes to prepare the people for future earthquakes.

Seismology

Voltage of Alternating Current

Modern power companies use alternating current to send electricity over long-distance wires. In an alternating current, the electrical charge regularly reverses direction to deliver power safely and reliably to homes and businesses. Electrical engineers use trigonometry to model this flow and the change of direction, with the sine function used to model voltage. Every time you flip on a light switch or turn on the television, you’re benefiting from one of trigonometry's many uses.

Voltage of Alternating Current

Temperature Cycles

Temperature

Cycles

Variables that depend on the seasons may be modeled with trigonometric functions because the seasons repeat every year just like the sine function repeats every 2π. Another neat application of trig curves (in particular, sine and cosine) is modeling temperature data.

Temperatures are cyclical. In the Northern Hemisphere, it tends to get cold in the winter and warmer in the summer. But why? In the southern hemisphere, our winter is their summer, even though the earth is closest to the sun in December.

GRAPHING

Graphs of

y = a sin x and y = a cos x

Graphs of

y = a sin x and

y = a cos x

y = 2 sin x

a. The function is odd.

b. Domain: R

c. Range: [-2 ,2]

d. Period: 2π

e. Amplitude: 2

f. No phase shifts

g. No movement in the resulting graph

Graphs of

y = a sin bx and y = a cos bx

Graphs of

y = a sin bx and y = a cos bx

y = cos 2x.

a. The function is even.

b. Domain: R

c. Range: [-1 ,1]

d. Period: 2π

e. Amplitude: 1

f. No phase shifts

g. No movement in the resulting graph

Graphs of

y = a sin b (x – c) + d and y = a cos b (x – c) + d

Graphs of

y = a sin b (x – c) + d

and

y = a cos b

(x – c) + d

y =2 sin (x + π/2).

a. The function is odd.

b. Domain: R

c. Range: [-2 ,2]

d. Period: 2π

e. Amplitude: 2

f. Phase shifts: π/2 units to the left

g. No movement in the resulting graph

Graphs of

Cosecant and Secant Function

Graphs of Cosecant and Secant Function

y = csc x

a. The function is odd.

b. Domain: {x/x R ,x ≠kπ;k Z}

c. Range: (-∞,-1] U [1,∞)

d. Period: 2π

e. Amplitude: n/a

f. Asymptotic at x=kπ

Graphs of

Tangent and Cotangent Function

y = tan x

a. The function is odd.

b. Domain: {x/x R ,x ≠π/2+ kπ;k Z}

c. Range: R

d. Period: π

e. Amplitude: n/a

f. Asymptotic at x=π/2 + kπ

Graphs of Tangent and Cotangent Function

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