WEEK 10

TRIGONOMETRIC FUNCTIONSTRIGONOMETRIC FUNCTIONS OF REAL

NUMBERS; PERIODIC FUNCTIONS

OBJECTIVES

Use a unit circle to define trigonometric functions of real numbers.Recognize the domain and range of sine and cosine functions.Use even and odd trigonometric functions.Use periodic properties.

INDEX1. Trigonometric functions of real numbers2. Finding values of trigonometric function3. Domain and range of sine and cosine functions4. Even and odd trigonometric functions5. Periodic functions6. Summary

So far we have studied trigonometric functions of angles measured in degrees or radians. In this section we will define trigonometric functions of real numbers, rather than angles.In order to define trigonometric functions of real functions we will use a unit circle.

The unit circle is the circle whose center is at the origin of a rectangular coordinate system and whose radius is one. The equation of this circle is The circumference of the unit circle is 2r= 22.

The figure shows a unit circle in which the central angles measures t radians. We use the formula for the length of a circular arc, s = r , to find the length of the intercepted arc.

s = r = 1. t = t (As radius of the unit circle is 1 and

radian measure of central angle is t radians)Hence the length of the intercepted arc is t which is same as the radian measure of the angle. Thus, in a unit circle, the radian measure of the angle is equal to the measure of the intercepted arc.

1. TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS

2 2x y 1

t

1

1 s = t

(1, 0)

Unit Circle

P = (x, y)

2 2x y 1

Let P = (x, y) denote the point on the unit circle that has arc length t from point (1, 0). Figure 1 shows that if t is positive, point P is reached by moving in a counter clockwise direction along the unit circle from (1, 0). Figure 2 shows that if t is negative, then the point P is reached by moving in a clockwise direction along the unit circle form point (1, 0). For each real number t, there corresponds a point P = (x, y) on the unit circle.As the radius of the unit circle is one, the trigonometric functions sine and cosine have special relevance for the unit circle. If a point P = (x, y) on the circle is on the terminal side of an angle t in standard position, then the sine of such an angle t is simply the y-coordinate of P, and the cosine of the angle t is the x-coordinate of P.Thus, x = cos t and y = sin t.

Example: If the point P = ( -2, -4). Then cos t = -2 and sin t = -4.Definitions of the Trigonometric Functions in Terms of a Unit Circle:If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then

As the definition expresses function values in terms of coordinates of a point on the unit circle, the trigonometric functions are also called the circular functions.

TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS(Cont…)

(1, 0)

1

t

1

s = t

(1, 0)

Unit Circle

P = (x, y)

2 2x y 1

Figure 2

t

1

1 s = t

Unit Circle

P = (x, y)

2 2x y 1 Figure 1

ysin t y cos t x tan t , x 0

x1 1 x

csc t , y 0 sec t ,x 0 cot t , y 0y x y

2. FINDING VALUES OF THE TRIGONOMETRIC FUNCTIONS

Example: Using the given figure we have to find the values of the trigonometric functions at t =

Solution: The coordinates of point P on the unit circle are (-1, 0). Thus we have x = -1 and y = 0.

From the definition of trigonometric functions in terms of unit circle, we have: sin = y = 0 (Substituting the value of y) cos = x = -1 (Substituting the value of x) tan = csc = sec =

cot =

Observe that the value of the trigonometric function at is its value at an angle of radians.

11 (1, 0)

Unit Circle

P = (-1, 0)

2 2x y 1

y 00

x 11 1

undefinedy 0

1 11

x 1x 1

undefinedy 0

3. DOMAIN AND RANGE OF SINE AND COSINE FUNCTIONS

From the previous example we have observed that the value of the trigonometric function at the real number t is its value at an angle of t radians.Recall:The domain of a function is the set of all possible input values. If the domain is not specified, we usually take it to be the set of all real numbers for which the function is defined. The range of a function is the set of all outputs or in other words, the set of all possible answers when the domain values are substituted for the input variable.The domain and range for each trigonometric function can be found from the definition of the unit circle. We will confine ourselves to study of domain and range of sine and cosine functions only. From the definition of trigonometric functions in terms of a unit circle, we know

sin t = y and cos t = xAs t can be the radian measure of any angle or equivalently it can be the length of any intercepted arc, thus the domain of the sine and cosine function is the set of all real numbers.Next we know that the circumference of a circle of unit radius is 2. If a moving point P starts from A and moves in the counter clockwise direction, then at the points A, B, A’, B’ and A, the arc lengths covered are t = 0, , , 3, and 2

2 2 respectively.

1

t

1 s = t

A(1, 0)

Unit Circle

P = (x, y)

2 2x y 1

B(0, 1)

A’(-1, 0)

B’(0, -1)

3. DOMAIN AND RANGE OF SINE AND COSINE FUNCTIONS(Cont…)

The coordinates of these points are: A (1, 0), B (0, 1), A’ (-1, 0), B’ (0, -1) and A (1, 0).

At A (1, 0), t = 0 cos 0 = 1 and sin 0 = 0 (Substituting the values of x and y)

At B (0, 1), t = cos = 0 and sin = 1 At A’ (-1, 0), t = cos = -1 and sin = 0 At B’ (0, -1), t = cos = 0 and sin = -1 At A (1, 0), t = cos = 1 and sin = 0

Thus, we have As x = cos t and y = sin t, we get

Since the maximum output value is 1 and the minimum output value is –1, the range is the interval [-1, 1].  Thus, Domain and Range of the Sine and Cosine FunctionsThe domain of the sine function and the cosine function is the set of all real numbers. The range of the sine function and cosine function is the set of real numbers from –1 to 1, inclusive.    

t

1 s = t

A(1, 0)

Unit Circle

P = (x, y)

2 2x y 1

B(0, 1)

A’(-1, 0)

B’(0, -1)

2

32

0

2

2

2

32

32

32

222

1 x 1 and 1 y 1

1 cos t 1 and 1 sin t 1

2

4. EVEN AND ODD TRIGONOMETRIC FUNCTIONS

Recall: Even functionsA function f(x) is an even function if f(-x) = f(x) for all x in the domain of f. The right side of the equation of an even function does not change is x is replaced with –x.Odd functionsA function f(x) is an odd function if f(-x) = - f(x) for all x in the domain of f. Every term in the right side of the equation of an odd function changes sign if x is replaced with –x. From the figure we know that Coordinates of point P are (x, y)By the definition of trigonometric functions, x = cos t and y = sin tThus, coordinates of point P are (cos t, sin t) Similarly, coordinates of point Q are (cos (-t), sin (-t))From the figure we observe that x-coordinates of P and Q are same as they are in the positive direction of x-axis.Thus, cos (-t) = cos t. this shows that cosine function is even.But in case of sine function we observe that the y-coordinates of P and Q arenegatives of each other. Thus, sin (-t) = -sin t, this shows that sine function is odd.

NOTE: This argument is valid regardless of the length of t. The arc may terminate in any of the four quadrants or on any axis, these results hold.

t

1

1 t

(1, 0)

Unit Circle

P = (x, y)

2 2x y 1

Q = (x, -y)

-t

Thus, using the unit circle definition of the trigonometric functions we get the following results:

Even and Odd Trigonometric functions: The cosine and secant functions are even.

cos (-t) = cos t sec (-t) = sec t The sine, cosecant, tangent and cotangent functions are odd.

sin (-t) = -sin t csc (-t) = -csc t tan (-t) = -tan t cot (-t) = -cot t

Using odd and even functions to exact values: Let us find the exact value of cos (-60°)

As cosine function is even, so cos (-t) = cos tThus, cos (-60°) = cos 60° = (As cos 60° = )

Next let us find the exact value of tan As we know that tangent function is odd, thus tan (-t) = -tan tHence, tan = - tan = (As tan = )

4. EVEN AND ODD TRIGONOMETRIC FUNCTIONS(Cont…)

12

12

- 6

- 6

6

- 3 3

6

3 3

If an angle corresponds to a point P (x, y) on the unit circle, it is not hard to see that the angle + 2 corresponds to the same point P(x, y). If we begin at point P on the unit circle and travel a distance of 2 units along the perimeter of the circle, we will return to the same point P. There are many cases in which more than one angle has the same value for its sine, cosine, or some other trigonometric functions. This phenomenon exists because all trigonometric functions are periodic.A periodic function is a function whose values (outputs) repeat in regular intervals. Symbolically, a periodic function is represented as : f(t + c) = f(t),for some constant c . The constant c is called the period - - it is the interval at which the function has a non - repeating pattern beforerepeating itself again.

Thus, we can define periodic functions as :Periodic Functions:A function f is periodic if there exists a positive number c such that

f(t + c) = f(t) for all t in the domain of f. The smallest number c for which f is periodic is called the period of f.

5. PERIODIC FUNCTIONS

P (x, y)

+ 2

X

Y

As all the trigonometric functions are defines as the coordinates of point P, thus we obtain the following results:

Periodic Properties of Sine and Cosine Functions:sin (t+2) = sin t and cos(t+2) = cos t.

The sine and cosine functions are periodic functions and have a period 2.

Like sine and cosine functions, the secant and cosecant functions also have a period of 2.But the tangent and cotangent functions have a period . That is, if we begin at a point P (x, y) on the unit circle and travel a distance of units along the perimeter, we arrive at a point (-x, -y). The tangent function defines in terms of coordinates of a point, is the same at(x. y) and (-x, -y).

=

Thus, we have tan(t + ) = tan t. Similarly, the periodic properties hold for cotangent function,

cot(t + ) = cot t

5. PERIODIC FUNCTIONS(Cont…)

P (x, y)

X

Y

(-x, - y)

xy

-x-y

Tangent function at (x, y)

Tangent function at (-x, -y)

Thus, periodic properties of tangent and cotangent functions can be summed up as:

Periodic Properties of Tangent and Cotangent Functions:tan (t + ) = tan t and cot (t + ) = cot t

The tangent and cotangent functions are periodic functions and have a period of .

Using periodic properties to find exact values: cos 405° = cos (45° + 360°) = cos 45° = (360° = 2)

sin = sin ( +2 ) = sin = (sin = cos = )

Repetitive behavior of the sine, cosine and tangent functions:As the trigonometric functions are defined in terms of the coordinates of a point P, if we add (or subtract) multiples of 2 , the trigonometric functions sine and cosine do not change.Similarly, the trigonometric values for the tangent and cotangent functions do not change if we add or subtract multiples of . Thus, we arrive at the following generalization:

For any integer n and real number t, sin (t + 2 n) = sin t cos (t + 2 n) = cos t tan (t + n) = tan t

5. PERIODIC FUNCTIONS(Cont…)

2 2

94

4

4

2 2

4

4

2 2

5. PERIODIC FUNCTIONS(Cont…)Example 1: Let us find the exact value of tan

Solution: From the repetitive behavior of tangent function we know that tan (t + n) = tan ttan = tan ( + 2 ) = tan = 3 ( n = 2)

Example 2: Now let us find the exact value of tan 420° Solution:

From the repetitive behavior of tangent function we know that tan (t + n) = tan ttan 420°= tan (60° + 2. 180 ° ) = tan 60° = 3 ( n = 2 and 180° = radians)

7 3

7 3

3

3

6. SUMMARYLet us recall what we have learnt so far:

Definitions of the Trigonometric Functions in Terms of a Unit Circle:

If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then

Domain and Range of the Sine and Cosine FunctionsThe domain of the sine function and the cosine function is the set of all real numbers. The range of the sine function and cosine function is the set of real numbers from –1 to 1, inclusive.    Even and Odd Trigonometric functions:The cosine and secant functions are even.

cos (-t) = cos t sec (-t) = sec tThe sine, cosecant, tangent and cotangent functions are odd.

sin (-t) = -sin t csc (-t) = -csc t tan (-t) = -tan t cot (-t) = -cot t

ysin t y cos t x tan t ,x 0

x1 1 x

csc t , y 0 sec t ,x 0 cot t , y 0y x y

6. SUMMARY(Cont…)Periodic Functions:A function f is periodic if there exists a positive number c such that

f(t + c) = f(t) for all t in the domain of f. The smallest number c for which f is periodic is called the period of f.Periodic Properties of Sine and Cosine Functions:

sin (t+2) = sin t and cos(t+2) = cos t.The sine and cosine functions are periodic functions and have a period 2.Periodic Properties of Tangent and Cotangent Functions:

tan (t + ) = tan t and cot (t + ) = cot tThe tangent and cotangent functions are periodic functions and have a period of .Repetitive behavior of the sine, cosine and tangent functions:For any integer n and real number t,

sin (t + 2 n) = sin t cos (t + 2 n) = cos t tan (t + n) = tan t