Graphs Of The Trigonometric Functions

Let us begin by introducing a few algebraic language. When we write "nπ," where n could be any integer, we mean "any multiple of π."

0, ±π, ±2π, ±3π, . . .

We would touch on the following:

Zeros of a function

The graph of y = sin x

The period of a function

The graph of y = cos x

The graph of y = sin ax

The graph of y = tan x

Question 1

Which numbers are shown by the following, where n could be any integer?

a) 2nπ

The even multiples of π:

0, ±2π, ±4π, ±6π, . . .

2n, in algebra, basically signifies an even number.

b) (2n + 1)π

The odd multiples of π: ±π, ±3π, ±5π, ±7π, . . .

2n + 1 (or 2n − 1) typically signifies an odd number.

Zeros

By the zeros of sin θ, we mean those values of θ for which sin θ will equal 0.

Now, where are the zeros of sin θ? That is,

sin θ = 0 when θ = ?

In unit circle the value of sin θ is equal to the y-coordinate. Thus, sin θ = 0 at θ = 0 and θ = π -- and at all angles coterminal with them. In other words,

sin θ = 0 when θ = nπ.

This will be true, moreover, for any argument of the sine function. For instance,

sin 2x = 0 when the argument 2x = nπ;

that is, when

X=nπ/2

Which numbers are these? The multiples of nπ/2

0, ±,π/2,, ±π, ±,3π/2, . . .

Problem 2. Where are the zeros of y = sin 3x?

At 3x = nπ; that is, at

x = nπ/3

Which numbers are these?

The multiples of π/3.

The graph of y = sin x

The zeros of y = sin x are at the multiples of π. And it is there that the graph crosses the x-axis, because there sin x = 0. But what is the maximum value of the graph, and what is its minimum value?

sin x has a maximum value of 1 at π/2, and a minimum value of −1 at 3π/2 -- and at all angles coterminal with them.

Here is the graph of y = sin x:

The height of the curve at every point is the line value of the sine. In the language of functions, y = sin x is an odd function. It is symmetrical with regard to the origin. 

sin (−x) = −sin x.

y = cos x is an even function.

The independent variable x is the radian measure. x may be any real

number. We may assume the unit circle rolled out, in both directions, along the x-axis. See Arc Length.

The period of a function

When the values of a function frequently repeat themselves, we say that the function is periodic. The values of sin θ at regular intervals repeat themselves

every 2π units. Hence, sin θ is periodic. Its period is 2π.

Definition.

If, for all values of x, the value of a function at x + p 

is equal to the value at x --

If f(x + p) = f(x)

-- then we say that the function is periodic and has period p.

The function y = sin x has period 2π, because

sin (x + 2π) = sin x.

The height of the graph at x is equal to the height at x + 2π -- for all x.

Problem 3.

a) In the function y = sin x, what is its domain?

Answer

x may be any real number. − < x < .

b) What is the range of y = sin x?

Answer

sin x has a minimum value of −1, and a maximum of +1.

−1 y 1

The graph of y = cos x

The graph of y = cos x is the graph of y = sin x shifted, or translated,π/2 units to the left.

For, sin (x + π/2) = cos x. The student familiar with the sum

formula can easily prove that.

On the contrary, it is possible to see directly that

Angle CBD is a right angle.

The graph of y = sin ax

Since the graph of y = sin x has period 2π, then the constant a in

y = sin ax

shows the number of periods in an interval of length 2π. (In y = sin x,a = 1.)

For instance, if a = 2 --

y = sin 2x

-- that means there are 2 periods in an interval of length 2π.

If a = 3 --

y = sin 3x

-- there are 3 periods in that interval:

While if a = ½ --

y = sin ½x

-- there is only half a period in that interval:

The constant a therefore shows how frequently the function oscillates; thus a lot of radians per unit of x.

(When the independent variable is the time t, as it frequently is in physics, then the constant is written as ω ("omega"): sin ωt. ω is called the angular frequency; a lot of radians per second.)

Question 4.

a) For which values of x are the zeros of y = sin mx?

Answer

At mx = nπ; that is, at x = nπ/m

b) What is the period of y = sin mx?

Answer

2π/m. Since there are m periods in 2π, then one period is 2π divided by m. Compare the graphs above.

Question 5

y = sin 2x.

a) What does the 2 show?

In an interval of length 2π, there are 2 periods.

b) What is the period of that function?

Answer

2π/2 = π

c) Where are its zeros?

Answer

At x = nπ/2

Question 6

y = sin 6x.

a) What does the 6 indicate?

In an interval of length 2π, there are 6 periods.

b) What is the period of that function?

Answer

2π/6= π/3

c) Where are its zeros?

Answer

At x = nπ/6

Question 7

y = sin ¼x.

a) What does ¼ show?

Answer

In an interval of length 2π, there is one fourth of a period.

b) What is the period of that function?

Answer

2π/¼ = 2π• 4 = 8π.

c) Where are its zeros?

Answer

At x = nπ/¼ = = 4nπ.

The graph of y = tan x

Here is one period of the graph of y = tan x:

Why is that the graph? Consider the line value DE of tan x in the 4th and 1st quadrants:

As radian x moves from − π/2 to π/2 , tan x takes on all real values. That is,

for

− π/2 < x < π/2,

− < tan x < .

Quadrants IV and I made up an entire period of y = tan x. In quadrant IV, tan x is negative; in quadrant I, it is positive; and tan 0 = 0. Again, here is the graph:

At the quadrantal angles − π/2 and π/2, , tan x does not exist. Thus the lines x = − π/2 and x = π/2 are vertical asymptotes.

Below is the complete graph of y = tan x.

The graph of Quadrants IV and I is repeated in Quadrant II (where tan x is negative) and quadrant III (where tan x is positive), and periodically along the whole x-axis.

Question

What is the period of y = tan x?

Answer

One period is from − π/2 to π/2 . Hence the period is the distance between those two points: π.

Graphs of Trigonometric Functions

We have provided below the graphs of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. On the $x$-axis are values of the angle in radians, and on the $y$-axis is f (x) , the value of the function at each given angle.

Figure %: Graphs of the six trigonometric functions

Convince yourself that the graphs of the functions are correct. Observe that the signs of the functions in reality precisely correspond with the signs diagrammed in the in Trigonometric Functions, and that the quadrantal angles follow the rules described.

Again, for instance, consider the definition of sine. Given a point on the terminal side of an angle, the sine of the angle is the ratio of the y-coordinate of that point to the distance between it and the origin. Now assume that angle altering, but the point remaining the same distance from the origin. The point traces the circumference of a circle. As the angle goes from 0 to radians, the y coordinate increases, and therefore the sine of the angle. As the angle goes from radians to Π radians, the y-coordinate decreases, and so does the sine of the angle, but everyone is still positive.

Then as the terminal side of the angle enters the third and fourth quadrant, the y coordinate of the point on the terminal side is negative, and first decreases, and then increases. All of these alterations are indeed reflected in the graph. Below in the figure the quadrants of the coordinate plane are shown in the graph.

The quadrants are represented in graphs in addition to the coordinate plane

Values of Trigonometric Functions for Common Angles

It is essential to know the values of the trigonometric functions for certain common angles. The values of the trigonometric functions for angles 0, 30o, 45o, 60o and 90o are provided in the table below.

Angle A 0 30o 45o 60o 90o

sin A 0 ½ 1/√2 √3/2 1

cos A 1 √3/2 1/√2 ½ 0

tan A 0 1/√3 1 √3 ∞

It is possible to get the values in the above table for 0, 45o and 90o through the use of the definitions from Trigonometry Module.

The values in the above table can be readily remembered with the use of the following mnemonic. The values of sin 0, sin 30o, sin 45o, sin 60o and sin 90o are simply given by the square roots of 0/4, 1/4, 2/4, 3/4 and 4/4. The values of cos 0, cos 30o, cos 45o, cos 60o and cos 90o are given by the square roots of 4/4, 3/4, 2/4, 1/4 and 0/4. Clearly, the values of tan for any angle are obtained by dividing the sine value by the cosine value.

Significant Trigonometric Identities

Some important trigonometric identities relating functions of a single angle (say, A) are given below.

sin2 A + cos2 A = 1 ... (1)

1 + tan2 A = sec2 A ... (2)

1+ cot2 A = cosec2 A ... (3)

These identities are essential in simplification and solution of problems. Their proofs are given below.

• Based on the right-angled triangle in the figure together with, sin A = a / b and cos A = c / b.

Thus sin2 A + cos2 A = (a2 + c2)/b2. 

By Pythagoras Theorem, a2 + c2 = b2 in a right-angled triangle. 

Thus sin2 A + cos2 A = 1 and equation (1) is proved.

• On dividing both sides of equation (1) by cos2 A, 

we get tan2 A + 1 = 1 / cos2 A = sec2 A. Equation (2) is proved.

• On dividing both sides of equation (1) by sin2 A, 

we get 1 + cot2 A = 1/sin2 A = cosec2 A. Equation (3) is proved. View more on psalmfresh.blogspot.com for more educating and tricks article Thank you