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Trigonometry Formulas Made Simple – A Step By Step Guide for Student

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Table of content
  1. What is Trigonometry?
  2. Trigonometry Formulas
    1. Periodicity Formulas
    2. Co-function Formulas
    3. Sum/Difference Formulas
    4. Double Angle Formulas
    5. Half Angle Formulas
    6. Product Formulas
    7. Sum to Product Formulas
    8. Pythagorean Formulas
    9. Pythagorean Formulas in Radical Form
    10. Odd-Even Formulas
    11. Product-to-Sum Formulas
    12. Sum-to-Product Formulas
  3. Trigonometric Values of Special Angles
  4. Trigonometric Identities
  5. Trigonometric Equations
  6. Why there is a need for Trigonometry Formula for Students?

Trigonometry Formulas

Most Trigonometry formulas revolve around ratios and extremely handy to solve complex problems in Trigonometry. If you want to appear for any competitive exams after your school then hands-on knowledge of different Trigonometry formulas is essential. The basic of any Trigonometry formula is a Trigonometry Identity. So, you must be curious to know about Trigonometric identities, let us discuss the same in the next section.

A. Periodicity
  • \(sin(x+2\pi )=sin\; x\)
  • \(cos(x+2\pi )=cos\; x\)
  • \(tan(x+\pi )=tan\; x\)
  • \(cot(x+\pi )=cot\; x\)
B.Co-function:
  • \(sin(90^{\circ}-x)=cos\; x\)
  • \(cos(90^{\circ}-x)=sin\; x\)
  • \(tan(90^{\circ}-x)=cot\; x\)
  • \(cot(90^{\circ}-x)=tan\; x\)
C. Sum/Difference:
  • \( \sin (x + y) = \sin(x) \cos(y) + \cos(x) \sin(y)\)
  • \(\cos(x + y) = \cos(x) \cos(y) – \sin(x) \sin(y)\)
  • \(\tan(x+y)=\frac{\tan\: x+\tan\: y}{1-\tan\: x\cdot \tan\: y}\)
  • \(\sin(x – y) = \sin(x) \cos(y) – \cos(x) \sin(y)\)
  • \(\cos(x – y) = \cos(x) \cos(y) + \sin(x) \sin(y)\)
  • \(\tan(x-y)=\frac{\tan\: x – \tan\: y}{1+\tan\: x\cdot tan\: y}\)
D. Double Angle:
  • \(\sin(2x) = 2\sin(x).\cos(x)\)
  • \(\cos(2x) = \cos^{2}(x) – \sin^{2}(x)\)
  • \(\cos(2x) = 2 \cos^{2}(x) -1\)
  • \(\cos(2x) = 1 – 2 \sin^{2}(x)\)
  • \(\tan(2x) = \frac{[2\: \tan(x)]}{[1 -\tan^{2}(x)]}\)
E. Half Angle:
  • \(\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}\)
  • \(\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}\)
  • \(\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\)

Also, \(\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))(1-\cos(x))}{(1+\cos(x))(1-\cos(x))}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{1-\cos^{2}(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{\sin^{2}(x)}}\\ \\ \\ =\frac{1-\cos(x)}{\sin(x)}\)

So, \(\tan(\frac{x}{2}) =\frac{1-\cos(x)}{\sin(x)}\)

F. Product:
  • \(\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(x-y)}{2}\)
  • \(\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(x-y)}{2}\)
  • \(\sin\: x\cdot \sin\:y=\frac{\cos(x+y)-\cos(x-y)}{2}\)
G. Sum to Product:
  • \(\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\)
  • \(\sin\: x-\sin\: y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\)
  • \(\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\)
  • \(\cos\: x-\cos\: y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\)<
H. Pythagorean

Sin2x + Cos2x = 1

1 + tan2x = sec2x

1 + cot2x = cosec2x

I). Pythagorean in Radical Form

sinx = ∓√1cos2x

tanx = ∓√sec2x-1

cosx = ∓√1sin2x

J). Odd-Even

Also called negative angle identities

Sin(-x)=-sin x

cos(-x)=-cos x

tan(-x)=-tan x

cot(-x)=-cot x

sec(-x)=-sec x

cosec(-x)=-cosec x

Ratio or Quotient Identities are given as:

Sinθ = Cosθ X Tanθ

Cosθ = Sinθ X Cotθ

Tanθ = Sinθ Cosθ

Cotθ = CosθSinθ

K. Product-to-Sum
Product-to-Sum Formulas
L. Sum-to-Product
Sum-to-Product Formulas

Trigonometric Values of Special Angles

Degreesincostancotseccosec
0∘010Not Defined1Not Defined
30∘\[\frac{1}{2}\]\[\frac{√3}{2}\]\[\frac{1}{√3}\]√3\[\frac{2}{√3}\]2
45∘\[\frac{1}{√2}\]\[\frac{1}{√2}\]11√2√2
60∘\[\frac{√3}{2}\]\[\frac{1}{2}\]√3\[\frac{1}{√3}\]2\[\frac{2}{√3}\]
90∘10Not Defined0Not Defined1

What is Trigonometry?

In mathematics, Trigonometry shows the relationship between multiple sides and angles of a triangle. Trigonometry is used throughout the geometry where shapes are broken down into a collection of triangles. Basically, Trigonometry is the study of triangles, angles, and different dimensions. Although the definition may sound simpler yet it is vital for modern engineering, complex mathematics study, architecture, logarithms, calculus, and other fields.

What is Trigonometry?

The word Trigonometry was derived from the Greek words triangle (trigōnon) and measure (matron) during the 16th century. So, this is not a new concept but into the existence of centuries and played an important role in the discovery of various mathematical and scientific theories. One of the biggest benefits of this mathematic technique was realized by the astronautical science and Indian astronomers.

Trigonometric Identities

Trigonometry identities are Trigonometric functions of one or more angles where equality is defined for both sides. The identities are used to solve any complex Trigonometric equations or expressions. One of the most popular applications of Trigonometric identities is the integration of non-trigonometric functions.

Trigonometric Identities

\(\sin \theta = \frac{Opposite}{Hypotenuse}\)

\(\sec \theta = \frac{Hypotenuse}{Adjacent}\)

\(\cos\theta = \frac{Adjacent}{Hypotenuse}\)

\(\tan \theta =\frac{Opposite}{Adjacent}\)

\(csc \theta = \frac{Hypotenuse}{Opposite}\)

\(cot \theta = \frac{Adjacent}{Opposite}\)

The Reciprocal Identities are given as:

\(cosec\theta =\frac{1}{\sin\theta }\)

\(sec\theta =\frac{1}{\cos\theta }\)

\(cot\theta =\frac{1}{\tan\theta }\)

\(sin\theta =\frac{1}{csc\theta }\)

\(cos\theta =\frac{1}{\sec\theta }\)

\(tan\theta =\frac{1}{cot\theta }\)

Trigonometric Equations

A Trigonometry equation is an expression that may hold true or false for any angle. If it holds true then it is a Trigonometry identity otherwise they are termed as conditional equations. These equations can be solved with the help of basic Trigonometric formulas and identities. There are only a few equations that can be solved manually otherwise you need a calculator and special skills to find the solution to any Trigonometry problem.

Why there is a need for Trigonometry Formula for Students?

As discussed earlier, Trigonometry is the study of triangles, angles, and different dimensions. Although the definition may sound simpler yet it is vital for modern engineering, complex mathematics study, architecture, and other fields.

With a deep understanding of Trigonometry, students would be able to work on precise angles of different sides of a triangle, calculation of distance among different triangle points and more important details that can be used for a variety of settings.

Further, Trigonometry skills may help you in exploring wider job options too including engineering, architecture, aeronautical study, etc. So, this is really important for the student to learn Trigonometry who are planning to enter the field of scientific science or engineering.

With the right mathematical skills, even the most complex problems can be understood in little time only without putting hard efforts. So, this would not be saying wrong that Trigonometry is one of the most valuable branches of mathematics and online programs can make learning even easier and more interesting to master the Trigonometry skills.

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