Sin Cos Tan Values

Trigonometry is the branch of mathematics that studies the relationships between angles and sides of triangles or that deal with angles, lengths, and heights of triangles and relations between different parts of circles and other geometrical figures. The Concepts of trigonometry are very useful in practical life as well as finds application in the field of engineering, astronomy, Physics, and architectural design.

These are in total 6 ratios that we study in trigonometry used to tell us about the triangle and Sine and Cos are two of them. We will discuss in detail about the Sin Cos ratio, formula and other concepts.

Trigonometric Ratios

In mathematics six Trigonometric Ratios for the right angle triangle are defined i.e Sine, Cosecant, Tangent, Cosecant, Secant respectively. These Trigonometric Ratios are real functions which relate an angle of a right-angled triangle to ratios of two of its side lengths. Sin and Cos are basic Trigonometric functions that tell about the shape of a right triangle.There are six Trigonometric Ratios, Sine, Cosine, Tangent, Cosecant, Secant and Cotangent and are abbreviated as Sin, Cos, Tan, Csc, Sec, Cot. These are referred to as ratios because they can be expressed in terms of the sides of a right-angled triangle for a specific angle \[\theta\].

• Adjacent: It is the side adjacent to the angle being taken for consideration.

• Opposite: It is the side opposite to angle being taken for consideration.

• Hypotenuse: It is the side opposite to the right angle of the triangle or the largest side of the triangle.

(Image will be uploaded soon)

• Sin θ = \[\frac{Perpendicular}{Hypotenuse}\] = \[\frac{Opposite}{Hypotenuse}\]

• Cos θ =  \[\frac{Base}{Hypotenuse}\] = \[\frac{Adjacent}{Hypotenuse}\]

• Tan θ = \[\frac{Perpendicular}{Base}\] = \[\frac{Opposite}{Adjacent}\]

• Cosec θ =\[\frac{Hypotenuse}{Perpendicular}\] = \[\frac{Hypotenuse}{Opposite}\]

• Sec θ = \[\frac{Hypotenuse}{Base}\] = \[\frac{Hypotenuse}{Adjacent}\]

• Cot θ = \[\frac{Base}{Perpendicular}\] = \[\frac{Adjacent}{Opposite}\]

Basic Identities of Sine and Cos

If \[A + B\] = \[90^{0}\],that is A and B are complementary to each other then:

• \[Sin (A) = Cos (B)\]

• \[Cos (A) = Sin (B)\]

If \[A + B\] = \[180^{0} \] then:

• \[Sin (A) = Sin (B)\]

• \[Cos (A) = - Cos (B)\]

     \[ Cos^{2}(A) + Sin^{2}(A) = 1\]

Double and Triple Angles

The Double or Triple Angle ratios can be converted to Single angle ratio of Sin and Cos using the below mentioned formulae:

• \[Sin2A = 2 SinA.CosA \]

• \[Cos2A = Cos^{2}A – Sin^{2}A = 2Cos^{2}– 1 = 1 − 2Sin^{2}A \]

• \[Sin3A = 3 SinA – 4 Sin^{3}A \]

• \[Cos3A = 4 Cos^{3}A – 3 CosA \]

• \[Sin^{3}A = 4 Cos^{3}A SinA – 4 CosA Sin^{3}A \]

• \[Cos4 A = Cos^{4}A – 6 Cos^{2}A Sin^{2}A + Sin^{4}A \]

• \[Sin^{2} A = \frac{(1 – Cos2A)}{2} \]

• \[Cos^{2} A = \frac{(1 + Cos2A)}{2} \]

Sum and Difference of Angles

The Sin and Cos ratio of sum and difference of two angles can be converted two product using below mentioned identities:

• Sin(A + B) = Sin(A) Cos(B) + Cos(A) Sin(B)

• Sin(A − B) = Sin(A) Cos(B) − Cos(A) Sin(B)

• Cos(A + B) = Cos(A) Cos(B) − Sin(A) Sin(B)

• Cos(A − B) = Cos(A) Cos(B) + Sin(A) Sin(B)

• Sin(A + B + C) = SinACosBCosC + CosASinBCosC + CosACosBSinC − SinASinBSinC

• Cos(A + B + C) = CosACosBCosC − SinASinBCosC − SinACosBSinC − SinACosBSinC − CosASinBSinC

• \[SinA + SinB = 2Sin \frac{(A + B)}{2}.Cos\frac{A - B}{2}\]

• \[SinA – SinB = 2Sin\frac{A - B}{2}.Cos\frac{A + B}{2}\]

• \[CosA + CosB = 2Cos \frac{(A + B)}{2}.Cos\frac{A − B}{2}\]

• \[CosA + CosB = −2 Sin \frac{(A + B)}{2}.Sin\frac{A − B}{2}\]

In order to remember the trigonometric ratios values follow the given below steps:

We can the values for Sine ratios,i.e., 0, \[ \frac{1}{2}\], \[ \frac{1}{\sqrt{2}}\], \[ \frac{\sqrt{3}}{2}\], and 1 for angles \[0^{\circ} \],\[30^{\circ} \],\[45^{\circ} \], \[60^{\circ} \]and \[90^{\circ} \] and the Cos ratio will follow the exact opposite pattern i.e. 0, \[ \frac{1}{2}\], \[ \frac{1}{\sqrt{2}}\], \[ \frac{\sqrt{3}}{2}\], and 1 at\[90^{\circ} \],\[60^{\circ} \],\[45^{\circ} \],\[30^{\circ} \] and\[0^{\circ} \]. While the Tan will be the ratio of Sin and Cos ratio.

The value of Cosec, Sec and Cot is exactly the reciprocal of Sin, Cos and Tan.

Values of Trigonometric Ratios at Various Angles:

Trigonometry is a branch of mathematics and a sub-branch in algebra concerned with the measurement of specific functions of angles and their application to calculations. An example of trigonometry which is easy to understand is that of what architects use to calculate any particular distances.

Algebra and trigonometry are two major branches of mathematics. Algebra involves the study of math with specific formulas, rules, equations, and other variables. Trigonometry deals only with the triangles and their measurements.

The Six Main Functions of an Angle that are Commonly Used in Trigonometry are 

• sine (sin), 

• cosine (cos), 

• tangent (tan), 

• cotangent (cot), 

• secant (sec), and 

• cosecant (csc).

An easy and simple way to learn and understand Trigonometry is by studying all the basics of trigonometric angles and formulas by writing them down in a separate notebook which will be really useful to revise them before exams. Make sure you understand and study all the entire right-angle triangle concepts well so that you might compare any problems with a triangle before you try to solve them. The main thumb rule to score well in trigonometry is to learn your Pythagoras theorem with a whole heart. Keeping the Sine rule and Cosine rule at your fingertips will help you solve any type of problem in the examination. Finally, list down all the important identities and formulas of trigonometry in your mind and revision notes as well, and be thorough. Remember to learn how to use the trigonometry table in the necessary place.