History of Trigonometry-:
Trigonometry deals with the study of angles, triangles and trigonometric functions. It is derived from the Greek words trigonon which means triangle and metria which means measure. So trigonometry basically deals with the triangle measurement. The term came into use in the 17th century when trigonometry was started as an analytic science but its real origins lie in the ancient Egyptian pyramids and Babylonian astronomy that date back to about 3000 BCE.
The founder of science of trigonometry is the Greek astronomer and mathematician Hipparchus of Nicaea in Bithynia. Regarding the six trigonometric ratios Aryabhatta discovered the sine(sin) and cosine(cosin). Muhammad ibn Musa alkkhwarizimi discovered the tangent(tan). Abu al-Wafa’Buzjani discovered the secant(sec), cotangent(cot) and cosecant(cosec).
Formulas used in the trigonometry-:
In trigonometry different types of problems can be solved by using different trigonometric formulas. When we learn about trigonometric formulas, we consider them for right angled triangles only. In right angled triangle we have 3 sides namely hypotenuse, perpendicular and base. The longest side of the triangle is called hypotenuse. The line opposite to hypotenuse is called perpendicular. The third side is called the base.
The lists of formulas are as follows-:
- Basic Formulas
- Reciprocal Identities
- Trigonometry Table
- Periodic Identities
- Co-function Identities
- Sum and Difference Identities
- Double Angled Identities
- Triple Angled Identities
- Half Angled Identities
- Product Identities
- Sum to Product Identities
- Inverse Trigonometry Formulas
Basic Formulas-:
By using a right-angled triangle as a reference, the trigonometric functions or identities are derived:
- sin θ = Opposite Side/Hypotenuse
- cos θ = Adjacent Side/Hypotenuse
- tan θ = Opposite Side/Adjacent Side
- sec θ = Hypotenuse/Adjacent Side
- cosec θ = Hypotenuse/Opposite Side
- cot θ = Adjacent Side/Opposite Side
Reciprocal Identities-:
The Reciprocal Identities are given as:
- cosec θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
- sin θ = 1/cosec θ
- cos θ = 1/sec θ
- tan θ = 1/cot θ
All these are taken from a right angled triangle. With the height and base side of the right triangle given, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.
Trigonometry table-:
Below is the table for trigonometry formulas for angles that are commonly used for solving problems.
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
Angles (In Radians) | 0° | π/6 | π/4 | π/3 | π/2 |
sin | √(0/4)=0 | √(1/4) =1/2 | √(2/4) =√(1/2) =1/√2 | √(3/4)=√3/2 | √(4/4) = 1 |
cos | √(4/4) =1 | √(3/4) =√3/2 | √(2/4) =√(1/2) =1/√2 | √(1/4) =1/2 | √(0/4)=0 |
tan= sin/cos | 0/1 = 0 | (1/2)/(√3/2) = (1/√3) | (1/√2)/(1/√2) = 1 | (√3/2)/(1/2) = √3 | 1/0 = ∞ |
cot=cos/sin | 1/0 = ∞ | (√3/2)/(1/2) =√3 | (1/√2)/(1/√2) = 1 | (√3/2)/(1/2) =1/√3 | 0/1 =0 |
cosec=1/sin | 1/0 = ∞ | 1/(1/2) =2 | 1/(1/√2) = √2 | 1/(√3/2) = 2/√3 | 1/1 = 1 |
sec = 1/cos | 1/1 = 1 | 1/(√3/2) = 2/√3 | 1/(1/√2) = √2 | 1/(1/2) = 2 | 1/0 = ∞ |
The trigonometry standard angle table is created by the following steps-
Steps to Create a Trigonometry Table:
Step 1:
Create a table with the top row listing the angles such as 0°, 30°, 45°, 60°, 90°, and write all trigonometric functions in the first column such as sin, cos, tan, cosec, sec, cot.
Step 2: Determine the value of sin.
To determine the values of sin, divide 0, 1, 2, 3, 4 by 4 under the root, respectively. See the example below.
To determine the value of sin 0°
√(0/4)=0
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
Step 3: Determine the value of cos.
The cos-value is the opposite angle of the sin angle. To determine the value of cos divide by 4 in the opposite sequence of sin. For example, divide 4 by 4 under the root to get the value of cos 0°. See the example below.
To determine the value of cos 0°
√(4/4)=1
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
Step 4: Determine the value of tan.
The tan is equal to sin divided by cos. tan = sin/cos. To determine the value of tan at 0° divide the value of sin at 0° by the value of cos at 0° See example below.
tan 0°= 0/1 = 0
Similarly, the table would be.
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
tan | 0 | 1/√3 | 1 | √3 | ∞ |
Step 5: Determine the value of cot.
The value of cot is equal to the reciprocal of tan. The value of cot at 0° will obtain by dividing 1 by the value of tan at 0°. So the value will be:
cot 0° = 1/0 = Infinite or Not Defined
Same way, the table for a cot is given below.
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
cot | ∞ | √3 | 1 | 1/√3 | 0 |
Step 6: Determine the value of cosec.
The value of cosec at 0° is the reciprocal of sin at 0°.
cosec 0°= 1/0 = Infinite or Not Defined
Same way, the table for cosec is given below.
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
cosec | ∞ | 2 | √2 | 2/√3 | 1 |
Step 7: Determine the value of sec.
The value of sec can be determined by all reciprocal values of cos. The value of sec on 0∘ is the opposite of cos on 0∘. So the value will be:
sec0∘=1/1=1
In the same way, the table for sec is given below.
Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° |
sec | 1 | 2/√3 | √2 | 2 | ∞ |
Trigonometry quadrant system-:
There is a very simple way to understand about the concept of quadrant system for trigonometry that is-: you have to remember a line “All School To College“,
- ‘All‘ refers to the first quadrant which means that in the first quadrant the value of all trigonometric ratios will be positive.
- ‘School‘ refers to the second quadrant which means the value of sin and 1/sin =cosec will be positive and remaining values of trigonometric ratios will be negative.
- ‘To‘ refers to the third quadrant which means that the value of tan and cot will be positive and remaining values of trigonometric ratios will be negative.
- ‘College’ refers to the fourth quadrant which means that the value of cos and 1/cos = sec will be positive and remaining values of trigonometric ratios .
Periodicity Identities-:
These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.
- sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
- sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
- sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A
- sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
- sin (π – A) = sin A & cos (π – A) = – cos A
- sin (π + A) = – sin A & cos (π + A) = – cos A
- sin (2π – A) = – sin A & cos (2π – A) = cos A
- sin (2π + A) = sin A & cos (2π + A) = cos A
All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.
Co-function Identities-:
The co-function or periodic identities can also be represented in degrees as:
- sin(90°−x) = cos x
- cos(90°−x) = sin x
- tan(90°−x) = cot x
- cot(90°−x) = tan x
- sec(90°−x) = csc x
- csc(90°−x) = sec x
Sum & Difference Identities
- sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
- cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
- tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
- sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
- cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
- tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)
Double Angle Identities
- sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan^{2} x)]
- cos(2x) = cos^{2}(x)–sin^{2}(x) = [(1-tan^{2} x)/(1+tan^{2} x)]
- cos(2x) = 2cos^{2}(x)−1 = 1–2sin^{2}(x)
- tan(2x) = [2tan(x)]/ [1−tan^{2}(x)]
- sec (2x) = sec^{2 }x/(2-sec^{2} x)
- cosec (2x) = (sec x. csc x)/2
Triple Angle Identities
- Sin 3x = 3sin x – 4sin^{3}x
- Cos 3x = 4cos^{3}x-3cos x
- Tan 3x = [3tanx-tan^{3}x]/[1-3tan^{2}x]
Half Angle Identities-:
The half of the angle X is presented through the below formulas-:
Product identities
- sinx⋅cosy=sin(x+y)+sin(x−y)2
- cosx⋅cosy=cos(x+y)+cos(x−y)2
- sinx⋅siny=cos(x−y)−cos(x+y)2
Sum to Product Identities
- sinx+siny=2sinx+y2cosx−y2
- sinx−siny=2cosx+y2sinx−y2
- cosx+cosy=2cosx+y2cosx−y2
- cosx−cosy=−2sinx+y2sinx−y2
Inverse Trigonometry Formulas
- sin^{-1} (–x) = – sin^{-1} x
- cos^{-1} (–x) = π – cos^{-1} x
- tan^{-1} (–x) = – tan^{-1} x
- cosec^{-1} (–x) = – cosec^{-1} x
- sec^{-1} (–x) = π – sec^{-1} x
- cot^{-1} (–x) = π – cot^{-1} x
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