Math
Trigonometry Formulas Made Simple – A Step By Step Guide for Student
Table of Contents
Table of content
 What is Trigonometry?
 Trigonometry Formulas
 Trigonometric Values of Special Angles
 Trigonometric Identities
 Trigonometric Equations
 Why there is a need for Trigonometry Formula for Students?
Trigonometry Formulas
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.Cofunction:
 \(sin(90^{\circ}x)=cos\; x\)
 \(cos(90^{\circ}x)=sin\; x\)
 \(tan(90^{\circ}x)=cot\; x\)
 \(cot(90^{\circ}x)=tan\; x\)
C. Sum/Difference:
 \( \sin (x + y) = \sin(x) \cos(y) + \cos(x) \sin(y)\)
 \(\cos(x + y) = \cos(x) \cos(y) – \sin(x) \sin(y)\)
 \(\tan(x+y)=\frac{\tan\: x+\tan\: y}{1\tan\: x\cdot \tan\: y}\)
 \(\sin(x – y) = \sin(x) \cos(y) – \cos(x) \sin(y)\)
 \(\cos(x – y) = \cos(x) \cos(y) + \sin(x) \sin(y)\)
 \(\tan(xy)=\frac{\tan\: x – \tan\: y}{1+\tan\: x\cdot tan\: y}\)
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:
 \(\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(xy)}{2}\)
 \(\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(xy)}{2}\)
 \(\sin\: x\cdot \sin\:y=\frac{\cos(x+y)\cos(xy)}{2}\)
G. Sum to Product:
 \(\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{xy}{2}\)
 \(\sin\: x\sin\: y=2\cos\frac{x+y}{2}\sin\frac{xy}{2}\)
 \(\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{xy}{2}\)
 \(\cos\: x\cos\: y=2\sin\frac{x+y}{2}\sin\frac{xy}{2}\)<
H. Pythagorean
Sin^{2}x + Cos^{2}x = 1
1 + tan^{2}x = sec^{2}x
1 + cot^{2}x = cosec^{2}x
I). Pythagorean in Radical Form
sinx = ∓√1–cos2x
tanx = ∓√sec2x1
cosx = ∓√1–sin2x
J). OddEven
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 = ^{Cosθ}⁄_{Sinθ}
K. ProducttoSum
L. SumtoProduct
Trigonometric Values of Special Angles
Degree  sin  cos  tan  cot  sec  cosec 
0∘  0  1  0  Not Defined  1  Not Defined 
30∘  \[\frac{1}{2}\]  \[\frac{√3}{2}\]  \[\frac{1}{√3}\]  √3  \[\frac{2}{√3}\]  2 
45∘  \[\frac{1}{√2}\]  \[\frac{1}{√2}\]  1  1  √2  √2 
60∘  \[\frac{√3}{2}\]  \[\frac{1}{2}\]  √3  \[\frac{1}{√3}\]  2  \[\frac{2}{√3}\] 
90∘  1  0  Not Defined  0  Not Defined  1 
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.
The word Trigonometry was derived from the Greek words triangle (trigōnon) and measure (matron) during the 16^{th} 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 nontrigonometric functions.
\(\sin \theta = \frac{Opposite}{Hypotenuse}\)
\(\sec \theta = \frac{Hypotenuse}{Adjacent}\)
\(\cos\theta = \frac{Adjacent}{Hypotenuse}\)
\(\tan \theta =\frac{Opposite}{Adjacent}\)
\(csc \theta = \frac{Hypotenuse}{Opposite}\)
\(cot \theta = \frac{Adjacent}{Opposite}\)
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.
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