WEEK 9 TRIGONOMETRIC FUNCTIONS RIGHT TRIANGLE TRIGONOMETRY.

WEEK 9

TRIGONOMETRIC FUNCTIONS RIGHT TRIANGLE TRIGONOMETRY

OBJECTIVESAt the end of this session , you will be able to:

Use right triangles to evaluate trigonometric functions. Find function values for 30°, 45°, and 60°. Recognize and use fundamental identities. Use equal cofunctions of complements. Use right triangle trigonometry to solve applied problems.

INDEX1. Six Trigonometric Functions2. Evaluating Trigonometric Functions3. Fundamental Identities4. Function Values for some Special Angles5. Trigonometric Functions and Complements6. Applications7. Summary

The word trigonometry means measurement of triangles. We begin our study of trigonometry by defining the six trigonometric functions. Recall that a right triangle is a triangle whose one angle measures 90°Let us take any acute angle AOB (Figure). We take a point P on ray OB and drop the perpendicular PQ on OA. We denote the angle POQ by Greek letter (Theta). Then, we have a right triangle POQ in which angle QOP = We note that in the triangle POQ, OQ is the side adjacent to angle and PQ is the side opposite to angle The side opposite the right angle is known as the hypotenuse.For an acute angle of the triangle POQ the ratio of the length of the side opposite angle divided by the length of hypotenuse is called the sine of the angle.For example the sine of the angle in the figure is the length of PQ divided by the length of PO. This fact is denoted by

NOTE: sin is an abbreviation for sine and it is not the product of sin and . The expression sin means sin(), where sine is the name of the function

and , the measure of an acute angle. It is correctly read as ‘sine of angle ‘.

1. SIX TRIGONOMETRIC FUNCTIONS

P

QO

B

A

Side adjacent to angle

Hypotenuse Side opposite

Length of side opposite angle PQsin

Length of hypotenuse OP

For an acute angle of the triangle POQ the ratio of the length of the side adjacent to angle divided by the length of hypotenuse is called the cosine of the angle. For example the cosine of the angle in the figure is the length of OQ divided by the length of OP. This fact is denoted by

For an acute angle of the right triangle POQ the ratio of the length of the side opposite angle divided by the

length of the adjacent side to angle is called the tangent of the angle. For example, for the angle of the triangle POQ in the figure, the tangent of the angle is PQ/OQ. This is denoted as

We abbreviate the two functions respectively as cos , and tan . Each of these abbreviations is written in lower case letters.

1. SIX TRIGONOMETRIC FUNCTIONS(Cont…)

P

QO

B

A


HypotenuseSide

opposite

Length of side adjacent to angle OQcosine

Length of hypotenuse OP

Length of side opposite angle PQtangent

Length of side adjacent to angle OQ

Besides sine, cosine, and the tangent of an angle , there are three other trigonometric

functions, namely cosecant (abbreviated as csc), secant (abbreviated as sec), and

cotangent (abbreviated as cot) of the angle . We define them as follows:

NOTE: Observe that csc, sec , and cot are reciprocals of sin , cos , and tan

respectively.


P

QO

B

A


Hypotenuse Side opposite Length of hypotenuse OP

cosecantLength of side opposite angle PQ

Length of hypotenuse OPsecant

Length of side adjacent to angle OQ

Length of side adjacent to angle OQcot angent

Length of side opposite angle PQ


Now let us summarize the six trigonometric functions in the following table:We assume that be an acute angle in a right triangle as shown in the figure. The length of the side opposite is a, the length of the side adjacent to is b, and the length of the hypotenuse is c. Name of the Trigonometric function

Abbreviated form

Definition of the trigonometric function

Sine sin = Opposite = a Hypotenuse c

Cosine cos = Adjacent = b Hypotenuse c

Tangent tan = Opposite = a Adjacent b

Cosecant csc = Hypotenuse = c Opposite a

Secant sec = Hypotenuse = c Adjacent b

cotangent cot = Adjacent = b Opposite a

CA

B

Length of the side adjacent to angle

Length of the hypotenuse

Length of the side opposite

c

a

b

NOTE: We take the side opposite to angle A as a, side opposite angle B as b, and side opposite angle C as c.


REMARKS: The figure shows three triangles of varying sizes. In each

of these triangles the measure of angle is the same.

Notice that all these three triangles have the same shape and the lengths of their corresponding sides are in the same ratio. In each case the tangent function has the same value.

Thus, the angle remains the same measure. The lengths of the sides of the triangle change but the ratio of the sides will remain the same as before.

Like tan , the other trigonometric functions for angle will have the same values regardless of the size of the triangle. However, the values of these functions will change if the angle itself changes.

In general, the trigonometric function values of depend only on the measure of the angle , and not on the size of the triangle.

a = 3

b = 2

tan = a/b

= 3/2

a = 6

b = 4

tan = a/b

= 6/4= 3/2

a =1.5

b = 1.5

tan = a/b

= 1.5/1= 3/2

2. EVALUATING TRIGONOMETRIC FUNCTIONS

Example: We have to find the value of each of the six trigonometric functions of in the figure. Solution Steps: In order to find the values of six trigonometric functions, first we need to find the

lengths of all the three sides of the triangle (a, b and c). Given: Side BC, a = 10 Side AB, c = 26 We need to find side AC, that is, b. We find this side using the

Pythagorean Theorem Recall that Pythagorean Theorem: The sum of the squares of the lengths of the sides of a right triangle equals the square of the length of the hypotenuse. That is, if the sides have lengths a and b, and the hypotenuse has length c, then Substituting the values, we get,

(10)2 + (b)2 = (26)2

100 + (b)2 = 676 (b)2 = 676 – 100 = 576 b = 576 = 24.

Now that we know the lengths of all the three sides of the triangle, we apply the definitions of the six trigonometric functions of .

AC

B

10 26

b

2 2 2a b c

Now that we know the lengths of all the three sides of the triangle, we apply the definitions of the six trigonometric functions of .

2. EVALUATING TRIGONOMETRIC FUNCTIONS(Cont…)

AC

B

10 26

24

NOTE: The values of csc , sec , and cot can be found by exchanging the numerator and the denominator of sin , cos , and tan respectively.

Length of side opposite anglesin

Length of hypotenuse

10 5

26 13

Length of side adjacent to anglecos


24 12

26 13

Length of side opposite angletan

Length of side adjacent to angle

10 5

24 12

Length of hypotenusecsc

Length of side opposite angle

26 13

10 5

Length of hypotenusesec


26 13

24 12

Length of side adjacent to anglecot


24 12

10 5

3. FUNDAMENTAL IDENTITESMany relationships exist among the six trigonometric functions. These relationships are described using trigonometric identities.For instance, we define

and

We observe that csc can also be defined as a reciprocal of sin . Thus, this relationship can be defines as a reciprocal identity,

Hence, we have the reciprocal identities:

Length of side opposite anglesin


Length of hypotenusecsc


1csc

sin

1sin

csc

1csc

sin

1cos

sec

1sec

cos

1tan

cot

1cot

tan

Next, we define

(Multiply and divide both sides by“length

of hypotenuse”)

(From the definitions of sin and cos)

Thus, we have the following Quotient identities:



Length of side opposite angle Length of hypotenusetan .

Length of hypotenuse Length of side adjacent to angle

1tan sin .

cossin

tancos

3. FUNDAMENTAL IDENTITES(Cont…)

sintan

cos

cos

cotsin

This is known as a Quotient Identity

3. FUNDAMENTAL IDENTITES(Cont…)If the values of sin and cos are known then we can find the value of each of the remaining four trigonometric functions using reciprocal identities and the quotient identities.

Example: Given sin = (2/3) and cos = (5/3), we have to find the values of the remaining four trigonometric functions.

Solution Steps: Step 1: Use Quotient Identity to find the value of tan .

(Rationalizing the denominator)

sintan

cos2

2 3 23 .35 5 5

3

2 5 2. 5.

55 5

3. FUNDAMENTAL IDENTITES(Cont…)Step 2: Now we have the values of sin , cos , and tan , next we find the values of the

remaining

three functions using reciprocal identities.

(sin = (2/3))

(cos = (5/3))


(tan = 2 )

5

1 1 3csc

2sin 23

1 1 3sec

cos 5 5

3

3 5 3 5.

55 5

1 1 5cot

2tan 25

3. FUNDAMENTAL IDENTITES(Cont…)PYTHAGOREAN IDENTITIES:Now we will find other relationships using the Pythagorean Theorem.In the right triangle BAC, we have,BC2 +AC 2 = AB2 (Using Pythagorean Theorem)That is, we have a2 + b2 = c2 (1)

(Dividing both sides by c2)

From the figure we observe, sin = a and cos = b , c c

Thus from the above equation we get,

(2)NOTE: For convenience, we will use the notation sin2 for (sin )2 and cos2 for (cos )2.

We rewrite equation 2 using this notation,

A C

B

b

c

a

2 2

2 2

2 2

a b+ = 1

c c

a b+ = 1

c c

2 2sin + cos 1

2 2sin + cos 1 (3)

3. FUNDAMENTAL IDENTITES(Cont…)Similarly, dividing both sides of equation 1 by b2, we get,

(Equation 1: a2 + b2 = c2)

(By definitions of tan and sec )

(4)

Again dividing both sides of equation 1 by c2, we get,

(5)

Relations 3, 4 and 5 found above are called Pythagorean Identities.

A C

B

b

c

a

2 2 2

2 2 2

2 22

2 2

a b c+ =

b b b

a c+ 1 =

b b

tan + 1 sec

2 21+tan sec

2 21+cot csc

3. FUNDAMENTAL IDENTITES(Cont…)Example: Given sin = (1/2) use the Trigonometric Identity to find the value of cos , where is an acute angle.We will use the Pythagorean Identity sin2 + cos2 = 1, to find the value of cos .Substituting the value of sin in the above identity we get,

2

2

2 22

2

2

2

2

cos

1 1 1( )cos

2 2 4

cos

cos (Subtracting from both sides)

cos

cos

3cos (As is an acute angle, so, cos is positive)

2

4. FUNCTION VALUES FOR SOME SPECIAL ANGLES

Now we will learn how to calculate the values of trigonometric functions for 30° or radians, 45° or radians, and 60° or radians angles which occur frequently in trigonometry.

TRIGONOMTERIC FUNCTIONS OF 45°We construct a triangle ABC, right angled at B, in which angle A= angle C = 45°. Thus, the triangle is isosceles, that is, it has two sides equal. Hence, BC = AB.Assume AB = BC = aThen AC2 = AB2 + BC2 = a2 + a2 = 2a2 (By Pythagorean Theorem) Thus, AC = a2.Remembering that in triangle ABC, angle C = 45°, we get

B C

A

45°

a

a

a2

side opposite to angle 45 a 1sin 45

hypotenuse a 2 2

side adjacent to angle 45 a 1cos45

hypotenuse a 2 2

side opposite angle 45 atan 45 1

side adjacent to angle 45 a

1 11; sec45 2; cot 45 1csc45 2

cos45 tan 45sin 45

6

4

3

4. FUNCTION VALUES FOR SOME SPECIAL ANGLES(Cont…)

TRIGONOMETRIC FUNCTIONS OF 60° We construct an equilateral triangle ABC (that is, a triangle in which all sides are equal). Assume that each side has length equal to 2. Observe that here we are considering angle A = 60°.Now we consider half of the equilateral triangle. We have a right triangle, with hypotenuse of length 2 and one side of length 1. The length of the third side be a, we find the length of this side using Pythagorean Theorem.

a2 + 1 = 22

a2 + 1 = 4

a2 = 4 – 1 = 3a = 3

Thus, we find the values of the trigonometric function as

B

A C

60°

30°2

1

3side opposite to angle 60 3

sin 60hypotenuse 2

side adjacent to angle 60 1cos60

hypotenuse 2

side opposite angle 60 3tan 60 3

side adjacent to angle 60 1

1 2 1 1 1 1 3 3csc60 ; sec60 2; co t 60 .

sin 60 cos60 tan 60 33 3 3 3

TRIGONOMETRIC FUNCTIONS OF 30° We will consider the same figure for finding the values of the trigonometric functions of 30 °.Note: Here we find the values of the trigonometric functions with respect to the acute angle B = 30°


NOTE: If a radical appears in the denominator, always rationalize the denominator.


A

C

60°

30°

2

3

side opposite to angle 30 1sin30

hypotenuse 2

side adjacent to angle 30 3cos30

hypotenuse 2

side opposite angle 30 1tan30

side adjacent to angle 30 3

1 3 3.

33 3

1

B

1 1 2 2 3 2 3 1csc 30 2; s c 30 . ; cot 30 3

sin 30 s 30 3 tan 303 3 3e

co


Let us now give the values of trigonometric functions of 30°, 45°, and 60° together in the form of the following table:

Remark: Looking at the above table we make the following observations: The greater is the value of , the greater is the value of sin. The greater is the value of , the smaller is the value of cos. tan is increasing with .

sin cos tan

30° 12

32

13

45° 12

12

1

60° 32

12

3

5. TRIGONOMETRIC FUNCTIONS AND COMPLEMENTS

Consider a right triangle ABC, right angled at C. Assume that angle A is . Now as the sum of the angles of a triangle is 180°, thus in a right triangle the sum of the two acute angle is 90°. Hence, the two acute angles of a right triangle are complements of each other.

So, if the degree measure of one acute angle is , then the measure of the other acute angle is (90° - ). Considering the acute angle , we already know that,

(1)

Now considering the acute angle (90° - ), the length of the opposite side is b, and the length of the adjacent side is a.

Because of this relationship, the sine and cosine are called cofunctions of each other. The name cosine is a shortened form of the phrase complement’s sine. Thus any pair of trigonometric functions f and g for which f()= g(90° - ) and g() = f(90° - ) are called cofunctions.

CA

B

c

a

b

This angle is (90° - )

asin

cb

cosc

bsin(90° - ) cos

ca

cos(90° - ) sinc

Using (1) above

Similarly we can show that the tangent and cotangent as well as secants and cosecants are also cofunctions of each other.

Hence, we get the following definition,

Cofunction Identities:

The value of a trigonomteric function of is equal to the cofunction of the complement of .

sin = cos(90° - ) cos = sin (90° - )

tan = cot (90° - ) cot = tan (90° - )

sec = csc (90° - ) csc = sec (90° - )

NOTE: If is in radians, replace 90 ° with

Example: Find a cofunction with the same value as the given expression.

(a) sin 46°

The value of the trigonometric function of is equal to the cofunction of the complement of , so we need to find the complement of each angle.

Cofunction of sin is cos (90° - )

5. TRIGONOMETRIC FUNCTIONS AND COMPLEMENTS(Cont…)

2

5. TRIGONOMETRIC FUNCTIONS AND COMPLEMENTS(Cont…)

Substituting the value of in this expression, we get, sin 46° = cos (90° - 46° ) = cos 44°

(b) cot

Cofunction of cot is

Substituting the value of , we get

12

tan( )2

cot tan( )12 2 12

6tan( )

12 125

tan( )12

6. APPLICATIONSMany times we are required to find the height of a tower, building, tree, distance of a ship from a light house, width of a river, etc. Though we cannot measure them easily, we can determine these by using knowledge of trigonometric functions.

Suppose we wish to determine the height of a tree without actually measuring it. We could stand on the ground at a point A at some distance, say 12m, from the foot B of the tree.

Suppose the measure of acute angle BAC is 30°. Then we find the height BC of the tree by using trigonometric functions.

Thus, we have been able to find the height of the tree using

trigonometric functions.

Whenever, an engineer faces problem in determining the

width of a river (or height of a tower etc.); which may not be easily possible to measure with a measuring tape, he imagines a big right triangle. One of the sides of this triangle is the line drawn across the river (or a line drawn vertically down to the ground from the top of the tower etc.) and such that any one of the other two sides of the triangle and one of the angles can be easily measured by using any surveying instrument.

Knowing a side and angle, the engineer can use his knowledge of trigonometric functions to calculate the unknown side, that is, the width of the river (or height of the tower) as done above.

Tre

e

30°

B A

C

12 m

BC 1tan30

AB 3AB 12

Hence, BC 4 3 m3 3

6. APPLICATIONS(Cont…)Before we proceed to solve problems based on applications of trigonometric functions, let us first define a few terms:

Line of sight: Suppose we are viewing an object standing on the ground. Clearly, the line of sight (or the line of vision) to the object is the line from our eyes to the object, we are viewing.

Angle of Elevation: If the object is above the horizontal level of the eyes(that is, it is above the eye-level), we have to turn our heads upwards to view the object. In this processour eyes move through an angle. Such an angle is called the angle of elevation of the object from our eyes.

Angle of Depression: Suppose a boy, standing on the roofa building, observes an object lying on the ground at some distance from the building. In this case, he has to move his head downwards to view the object. In this processhis eyes again move through an angle. Such an angle is called the angle of depression of the object from the location of his eyes.

Object

Line of sight

Horizontal Line

Angle of Elevation

Object

Line of sight

Angle of Depression

Horizontal Line

6. APPLICATIONS(Cont…)Example: A flagpole that is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation of the sun to the nearest degree.Given that the height of the flagpole is 14 m and the length of its shadow is 10 m. We are asked to find angle . In order to find angle , we begin with the tangent function.From the definition of tangent function we know,

Substituting the values we get,

Using a calculator, we find that the value of the angle 54°Thus, the angle of elevation is 54° approximately.

14 m

10 m



14tan

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

There are six trigonometric functions which an be defined as

Reciprocal Identities:

Quotient identities:

Opposite Hypotenusesin = ; csc =

Hypotenuse Opposite

Adjacent Hypotenusecos = ; sec =

Hypotenuse Adjacent

Opposite Adjacenttan = ; cot =

Adjacent Op

posite

1sin

csc

1csc

sin

1cos

sec

1sec

cos

1tan

cot

1cot

tan

sintan

cos

cos

cotsin

7. SUMMARY(Cont…)Pythagorean Identities:

Cofunction Identities:

sin = cos(90° - ) cos = sin (90° - )

sec = csc (90° - ) csc = sec (90° - )

tan = cot (90° - ) cot = tan (90° - )

2 2sin + cos 1 2 21+tan sec 2 21+cot csc

WEEK 9 TRIGONOMETRIC FUNCTIONS RIGHT TRIANGLE TRIGONOMETRY.

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