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Trigonometrical Ratios of Standard Angles

2019-06-17T21:03:00.000-07:00

Trigonometrical Ratios of Standard Angles

We shall study trigonometrical ratios of some standard angles for other angles, and we can also use trigonometrical table.

Trigonometrical Ratio of 0° and 90° :

Let consider an acute angle ∠ABC = 𝜭 and let P be any point on its terminal side BC. Draw perpendicular PQ from P on BA.

In ⃤ BPQ, we have

In ⃤ AMP, If 𝜃 becomes smaller and smaller line segment PQ also becames smaller and smaller; and Finally when 𝜃 becomes 0°; the point P will parallel with Q.

PQ = 0 and BP = BQ

Thus, we have

Similarly as above as and on using reciprocal relation,

From ⃤ BPQ, it is obvious that as 𝜃 increase, line segment BQ becomes smaller and smaller and finally when 𝜭 became 90° the point Q will parallel with B.
Consequently, we have

PQ=0 and BP = PQ

Thus, we have

Sin90° = 1, cos90° = 0, tan90° = ∞, cot90° = 0, sec90° = ∞ ,and csc90° = 0

Trigonometrical ratio of 30° and 60° :

Let consider an equilateral triangle PQR with each side of length 2a. since each angle of an equilateral triangle is of 60° . therefore, each angle of ⃤ PQR is of 60°. Let PS be perpendicular from P and QR. Since the triangle is equilateral.

Therefore, PS is the bisector of ∠P and ∠S the mid-point of QR.

∴ QS = SR = a and ∠QPS = 30°

Thus, in ⃤ PQS, ∠S is a right angle, hypotenuse PQ = 2a and QS = a.

Applying pythagoros theorem, in ⃤ PQS, we obtain

PQ² = PS² + QS²

(2a)² = PS² + a²

PS² = 4a² - a²

PS = √3a

Trigonometrical ratios of 30° :

In ◿ PSQ, we obtain Base = PS = √3a,

Perpendicular = QS = a,

Hypotenuse = PQ = 2a, and

∠ SPQ = 30°

Trigonometrical ratios of 60° :

In◿PSQ, we have

Base = QS = a

Perpendicular = PS √3a

Hypotenuse = PS = 2a, and

∠PQS = 60°

Trigonometrical ratios of 45° :

Consider a right angle triangle ◿ PQR with right angle at R such that ∠P = 45°. Then,

∠P + ∠Q +∠R = 180°

⇒ 45° + 90° + 45° = 180°

⇒ ∠Q = 45°

∴ ∠P = ∠R

⇒ PR = RQ

Let PR² = RQ² = a . Applying pythagoros theorem, in ⊿PQR ,

we obtain

PQ² = PR² + RQ² PQ² = a² + a² PQ² = 2a² PQ = √2a Thus, in PQR, we have P = 45, Base = PR = a, Perpendicular = RQ = a, and Hypotenuse = PQ = √2a

PQ² = a² + a²

PQ² = 2a²

PQ = √2a

Thus, in PQR, we have

∠P = 45°, Base = PR = a, Perpendicular = RQ = a, and Hypotenuse = PQ = √2a

Trigonometric Table For Standard Angle :

***Trigonomeric Table***
𝜭	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	∞
csc	∞	2	2	√2/3	1
sec	1	2/√3	√2	2	∞
cot	∞	√3	1	1/√3	0

Trigonometric Ratios of Complementary Angles :

In this section, we will obtain the trigonometric ratios of complementary angles in terms of the trigonometric ratios of the given angles.

Complementary Angle ⇒

When sum of two angles are in 90° then it is called Complementary Angle.

from the above definition, If 𝜭 and (90 - 𝜭 ) are complementary angles for an acute angle 𝜭.

Theorem :

If 𝜽 is an acute angle, than prove that

sin (90 - 𝜽) = cos 𝜽,

cos (90 - 𝜽) = sin 𝜽,

tan (90 - 𝜽) = cot 𝜽,

cot (90 - 𝜽) = tan 𝜽,

sec (90 - 𝜽) = csc 𝜽, and

csc (90 - 𝜽) = sec 𝜽

Proof :

consider a right triangle ⊿ OAB, right angled at B as shown in fig Let ∠BOA = 𝜽, then ∠OAB = (90 - 𝜽).

For the reference angle 𝜽, we have

For the reference angle (90° - 𝜽), we have

Base = AB, Perpendicular = OB, and Hypotenuse = OA

By solving Both, we get

Trigonometric Ratios

2019-04-27T19:23:00.001-07:00

Trigonometric Ratios

Introduction:

"Trigonometry" is an important branch of mathematics. The word 'Trigonometry' made by two Greek words; trigonon and metron. The word trigonon means a triangle, the word metron means measure. So that we known that Trigonometry is the way were we introduced with the science of measuring an triangles. In 3rd century BC.

Concept of Perpendicular (height), Base, and Hypotenuse in a Right Angle Triangle:

For any acute angle in a right-angled triangle; Perpendicular is the side opposite to the acute angle; the side adjacent to it is called Base; Hypotenuse is the side opposite to its right angle.

Example:

Angle :

Let's Consider a BA ray. If this ray rotates through its origin B and takes the position BC, then we say that the angle ∠ABC has been generated.

The initial position AB is called the initial side of the angle and the final position BC is called the terminal side of the angle.Point B about which the ray rotated is called the vertex of the angle.

Notation of Angles :

To present an angle in a triangle we often use English alphabet, but in trigonometry we generally use the following Greek letters which are given below:

𝜽 (Theta)
𝞪 (Alpha)
𝞫 (Beta)
𝜸 (Gamma)
𝟇 (Phi)
𝜳 (psi)

etc.

Trigonometrical Ratios:

The ratio between the lengths of two sides of a right-angled triangle is called a Trigonometrical Ratios.
In a right angled triangle there was three sides which give us six trigonometrical ratios.

Which are:

Sine → Sin,
Cosine → Cos,
Tangent → Tan,
Cotangent → Cot,
Secant → Sec, and
Cosecant → Cosec.

In a right-angled triangle ABC , for an acute angle 𝜽 :

Consider an acute angle ∠ABC = 𝜽 with initial side BA. Let P be any point on the on the terminal side BC. Draw PQ perpendicular from P on BA to get the right angled triangle AQP in which ∠ PBQ = 𝜽 .

In the right angled triangle AQP, we have

Base = BQ = x, perpendicular = PQ = y, and

Hypotenuse = BP = r.

) Sine (sin): The ratio between the lengths of perpendicular and hypotenuse are called sine (sin).

Cosine (cos): The ratio between the lengths of base and hypotenuse are called Cosine (cos).

Tangent (tan): The ratio between the lengths of perpendicular and base are called Tangent(tan).

Cotangent (cot): The ratio between the length of base and perpendicular are called Cotangent (cot).

Secant (sec): The ratio between the length of hypotenuse and base are called Secant(sec).

Cosecant (cosec): The ratio between the length of hypotenuse and perpendicular are called Cosecant (cosec).

Note: sin𝛳 is a abbreviation of 'sine of angle 𝛳 ' similar to all case of other trigonometric ratios.
Tips 1): All trigonometric ratios are positive real number.
Tips 2): The above trigonometric ratios are defined for an acute angle 𝛳.

Similarly, for acute angle 𝟇 in the given right angled triangle.

Relations Between Trigonometric Ratios:

For an acute angle 𝛳. sin𝛳, cos𝛳, tan𝛳 are closely connected by an relation through trigonometry ratios. if any of them is known, the other two can be easily calculated.

From fig, We have

, , and

Now,

Reciprocal Relations:

sin 𝛳 and csc 𝛳 are reciprocal of each other.

Logarithm | log rules, formula and examples

2019-04-22T21:29:00.000-07:00

Logarithm

Introduction

Logarithm was discovered in 17th century in Europe as to simplify an mathematical calculations . It was a Greek word which means arithmetic, ratio, and proportion. If someone asked us, "multiply 10'times 2" what would do you simply multiple the 2 Ten times like that, 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 = 1,024. If I said that write it in another way so what you would do you us exponential expression by these you can write it as

2¹⁰= 1024

If I said that there is another way to solve or write this equation, what is the method? It was Logarithm an another way of thinking to solve exponential equations.

What is Logarithm ?

In previous classes we study exponential, we known that if 6'raised to the power 3 the resultant is 216. This is known as exponential expression. written as,

6³= 216

Let assume that someone asked us that if 6'raised to which power to get 216? . The answer was 3. This is expressed by the logarithmic equation.written as,

log₆216 = 3

we can read this equation as log of 216 at base 6 is 3, or log base 6 of 216 is 3.
Relation ship between exponential and logarithm expression.

6³= 216 ⇔ log_a216 = 3

6 is the Base
3 is the power(index - exponent)
216 is the resultant or argument

Use of logarithm

Logarithm is a revers exponent form because of this reason, they are very useful and helpful for solving and understanding exponential equations. Sometimes we come across Logarithm expression is very interesting to learn and understand, because it expresses numerical expressions involving multiplication, division, and rational power of large numbers. Logarithm are very helpful in making lengthy and difficult calculations easy. It is very common around us in the world. Here we introduce the concept of logarithms, and apply the technique to simplify numerical expressions.

Definition

Assume that a is a positive real number, other then 1(a≠1) and x is a rational number, then the exponential expression is a^x = n , So that we says that logarithm of n at base a is x or x is the log of n to the base a, written as log_an = x.
Thus,

a^x= n ⇔ log_an = x

log_an = x is a exponential expression, and
a^x = n is a logarithmic expression

Examples from above definition:

Remember that : Any positive real number a, we know that and .

∴ and .

Decimal Logarithm :

All of our examples we have used whole numbers like 2, 3,or 5 etc. but log have decimal values which we discussed in our previous examples (5.) where, we have seen that log 2 at base 10 is 0.3010.

for example :

Some other examples of the above definition that:

1.) Find the real number x ?
from the above definition that :
A real number x ⇔
Thus,

x=5
So that,
A real number is 5
2.) Find the real number x ?
from the above definition that :
A real number x ⇔
Thus,

x = 1/2

So that,
A real number is 1/2

3.) Find the real number x ?
from the above definition that :
A real number x ⇔
Thus,

x=-3
So that,
A real number is -3

4.) Find the real number x ?
from the above definition that :
A real number x ⇔
Thus,

x=1
So that,
A real number is 1
Remember that : "log" is the abbreviation of the word "logarithm"

Laws of Logarithm

The basic rules of logarithm can expressed in many different ways are collectively named as "fundamental Laws of logarithm" .
There are Four fundamental fundamental of logarithm .

1st law : Product Rule,

2nd law : Quotient Rule, and

3rd law : Power Rule.

Product Rule :

The logarithm of the product two numbers or more then two numbers is equal to the sum of their logarithms .

Proof :

If m,n are positive real numbers, then the product of them is (mn)

Let

and . Then,

[by definition of logarithm]

[by law of exponent]

[by definition of log]

[]

Quotient Rule :

The logarithm of the ratio of two numbers or more then two numbers is equal to the difference of their logarithms.

Proof :

If m and n are positive rational numbers, then,

[by definition of log]

[by exponential law]

[by the definition of log]

Note : if x, y are real numbers, (where xy > 0) and a is a positive real numbers (where a ≠ 1), then

Power Rule :

If m,n are positive rational number, and a is a positive real number (where a > 0 and a ≠ 1),

Proof :

[by definition of logarithm]

Some other Rules of Logarithm :

There are six other rules of logarithm .

First Law :

If a,b are positive real number (where a ≠ 1, b ≠ 1), and m is positive rational number, then

Proof :

[Taking log to the base b]

Note :

On replacing b by m in the above equation result, we get

Second law :

The log of 1 to any base is always Zero.

Proof :

We have, (where a > 0)

[Therefore, by definition of log]

Third Law :

The log of any positive number to the same base is always one.

Proof :

For any positive number a,

We know that,

Therefore, by definition of log,

Fourth Law :

If 'n' is a positive rational number, and 'a' is a positive real number, other then 1, Then

Proof :

Fifth Law :

If 'n' is a positive rational number, 'a' is a positive real number, (Where a ≠ 1), Then

Proof :

[Substituting the value of x and y]

Replacing a = n in the above result, we obtain

Six Law :

If x'a' is a positive real number, (Where a ≠ 1) and x,y are positive rational number, Then

Proof :

We know that :

Special log properties :

Various types of logarithms base, but here we will discuss about two only which are widely used in logarithm. Most of the calculation is done in only this two.

Common Logarithm :

A log whose base is 10 (x log at base 10 ) is called common logarithm . When we write the equation, we ignore the base of logarithm.

for example :

Natural logarithm :

A log whose base is e (x log at base e ) is called natural log. The value of e = 2.718282, where e is called Euler's number. In engineering it is written In(x).

There was always a confusion around us about Common log and Natural log. Because in mathematics "log" is used while in engineering "In" is used. this can generates confusion around students.
For example :

log (86)

In engineering students thinks that it is a (log 86 at base 10) and mathematician thinks (log 86 at base e ) this type of confusion around students.

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Trigonometrical Ratios of Standard Angles

Trigonometrical Ratios of Standard Angles

Trigonometrical Ratio of 0° and 90° :

Trigonometrical ratio of 30° and 60° :

Trigonometrical ratios of 30° :

In ◿ PSQ, we obtain Base = PS = √3a, Perpendicular = QS = a, Hypotenuse = PQ = 2a, and ∠ SPQ = 30°

Trigonometrical ratios of 60° :

Trigonometrical ratios of 45° :

Trigonometric Table For Standard Angle :

Trigonometric Ratios of Complementary Angles :

Complementary Angle ⇒

Theorem :

Proof :

Trigonometric Ratios

Trigonometric Ratios

Introduction:

Concept of Perpendicular (height), Base, and Hypotenuse in a Right Angle Triangle:

Example:

Angle :

Notation of Angles :

Trigonometrical Ratios:

Which are:

Note: sin𝛳 is a abbreviation of 'sine of angle 𝛳 ' similar to all case of other trigonometric ratios.Tips 1): All trigonometric ratios are positive real number.Tips 2): The above trigonometric ratios are defined for an acute angle 𝛳.

Relations Between Trigonometric Ratios:

Reciprocal Relations:

Logarithm | log rules, formula and examples

Logarithm

Introduction

What is Logarithm ?

Use of logarithm

Definition

Examples from above definition:

Decimal Logarithm :

Some other examples of the above definition that:

Laws of Logarithm

Product Rule :

Proof :

Quotient Rule :

Proof :

Power Rule :

Proof :

Some other Rules of Logarithm :

First Law :

Proof :

Second law :

Proof :

We have, (where a > 0)

Third Law :

Proof :

For any positive number a,

Fourth Law :

Proof :

Fifth Law :

Proof :

Six Law :

Proof :

Special log properties :

Common Logarithm :

Natural logarithm :

In ◿ PSQ, we obtain Base = PS = √3a,

Perpendicular = QS = a,

Hypotenuse = PQ = 2a, and

∠ SPQ = 30°

Note: sin𝛳 is a abbreviation of 'sine of angle 𝛳 ' similar to all case of other trigonometric ratios.
Tips 1): All trigonometric ratios are positive real number.
Tips 2): The above trigonometric ratios are defined for an acute angle 𝛳.