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Trigonometriy - I

         Trigonometry is the branch of Mathematics that deals with measurement of angles. Three angles at a time are found in a triangle. A triangle is either scalene or isosceles or equilateral or right angled or isosceles right angled.
          of the triangles mentioned above, right angled triangle is considered for defining T-ratios such as sinθ, cosθ, tanθ, cotθ, secθ and cosecθ.
* The circular functions sinθ, cosθ, tanθ are significant in being termed as periodic functions, their periods being 2Π, 2Π, Π respectively.
* Sin (ax+b), cos (ax+b), tan (ax+b) possess periods respectively given as 
* The period of sin (x + 2x + 3x + . . . + nx)  
* The period of tan (x + 2x + 3x + . . . + nx)  


Pairs of results:
* sinθ sin (60°- θ) sin (60°+ θ) = 1/4 sin3θ
* cosθ cos (60°- θ) cos (60°+ θ) = 1/4 cos3θ.    
*  i) (Cosα + Cosβ)2 + (Sinα+ Sinβ)2 = 4Cos2.
*  ii) (Cosα - Cosβ)+ (Sinα - Sinβ)2 = 4 Sin2.
*  If α, β are the solutions of the equation a Cosθ + b Sinθ = c (a, b, c are non-zero real numbers) then
*  sin5 θ  = 16 sin5 θ - 20 sin3 θ+ 5 sinθ
* cos5 θ = 16 cos5 θ - 20 cos3 θ + 5 cosθ


Similar type problems in Pairs:

Trigonometric Equations

An equation involving trigonometric functions as variables is called a Trigonometric equation. Trigonometric functions are periodic and so they admit infinite number of solutions. The set containing all the solutions is called the general solution.
          Solutions of standard trigonometric equations.

          Whenever the terms of two sides of an equation are of different nature, then the equations are said to be in non-standard form. Some of them are in ordinary form, but can be solved by standard technique; on the other side, non standard problems require high degree of logic which can be solved by using graphs, inverse property of functions, and inequalities like

Number of integral values of K is 8 i.e, K = -4, -3, -2, -1, 0, 1, 2, 3.
Inverse Circular Functions: Trigonometric functions in general are not bijective. But by restricting their domains and codomains they can be made bijective so that their inverse exists. 

eg.: Find the AM of non zero solutions of equations

Properties of Triangles

Using trigonometry we can determine the relation between the sides and angles of a triangle. If one or two sides and one or two angles are given we can determine the remaining sides and remaining angles of the triangle. Two important rules widely applied are Sine Rule and Cosine Rule.
In triangle ABC, the sides BC, CA and AB are denoted with a,b and c respectively. If a circle is drawn passing through A, B, C then It is called the circum circle of ΔABC and 'O' is called the circumcentre. OA = R is the circum radius. The sine rule is given by
If one side and two angles are given using the above sine rule we can solve the traingle. That is we can find the other sides and angles. This is used to convert sides into angles or angles into sides to simplify the expressions.
Suppose if sides a and b are given and angle C is given, then we cannot apply sine rule.


We use cosine rule which is defined as c = a2+ b2 - 2ab cosc and the other two expressions
are also defined as a2 = b2+ c- 2bc cos A and b2 = a2 + c2 - 2ac cos B.
     These two rules are also called as Law of Sines and Law of Cosines.

These three results are called 'Mollewide' rules.

When all the three sides of the triangle a, b, c are given, by using Law of Cosines we can find the angles of the triangle.
eg.: If a = 6, b = 5, c = 9 then find the angle  of the triangle Δ ABC.

       Similarly, we can find the other two angles also.

*  The following three results are called Napier's Analogy or Tangent Rules:
*  The following three relations are called 'Projection Rules'. They can be proved using 'Law of cosines'.
                  In ΔABC, a = b cosC + c cos B

Similarly, we can show that b = c cosA + a cosC and c = a cosB + b cosA.
       If a,b,c are three sides of a ΔABC, then (a + b + c) is denoted by 2s and we have the following results called Half-Angle formulae.
* From the above results we can have the following

Using the above results we can express the area of the triangle Δ, in many ways.
1) Δ = area of ΔABC 
2) Δ = 2R sinA. sinB. sinC

Posted Date : 07-11-2020

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.

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