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solniwko [45]
3 years ago
7

Prove that if sin (2 + y) = 3 sin (x - y), then tan x = 2 tany.

Mathematics
1 answer:
nignag [31]3 years ago
7 0

Answer:

The proof is given below.

Step-by-step explanation:

Given: ( the correct question and ans is as follow )

sin (x + y) = 3sin ( x - y)

To Prove

tan x = 2 tan y

Proof:

\sin (x+y) = 3\sin (x-y)

Using the identities

\sin (A+B) = \sin A.\cos B + \cos A.\sin B\\and\\\sin (A-B) = \sin A.\cos B - \cos A.\sin B\\

we get

\sin x.\cos y + \cos x.\sin y = 3(\sin x.\cos y - \cos x.\sin y)\\\sin x.\cos y + \cos x.\sin y = 3\sin x.\cos y - 3\cos x.\sin y

Now we will take sin x . cos y to the left hand side and cos x . sin y to the right hand side,we get

3\sin x.\cos y - \sin x.\cos y = 3\cos x.\sin y +\cos x.\sin y\\2\sin x.\cos y =4\cos x.\sin y\\\sin x.\cos y =2\cos x.\sin y \ \textrm{4 divide by 2}\\ \frac{\sin x}{\cos x} =2\times \frac{\sin y}{\cos y} \\\textrm{we know identity }\\\frac{\sin x}{\cos x} = \tan x\\\frac{\sin y}{\cos y} = \tan y\\\therefore \tan x = 2\tan y\ ...............{PROVED}

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<h3>Further explanation</h3>
  • The angle is determined by three points, we call it A, B, C, with A ≠ C and B ≠ C.  
  • We express an angle with three points and a symbol ∠. The middle point represents constantly vertex. We can, besides, give angle names only with vertices. For example, based on the accompanying image, the angle can be symbolized as ∠BAC, or ∠CAB, or ∠A.  

<u>Note:  </u>

This is one of the classic problems of Euclidean geometry.

Undefined terms are the basic figure that is undefined in terms of other figures. The undefined terms (or primitive terms) in geometry are a point, line, and plane. These key terms cannot be mathematically defined using other known words.

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  • A plane represents a planar surface that contains many points and lines. A plane extends infinitely in all four directions. It is two-dimensional. Three noncollinear points determine a plane, as there is exactly one plane that can go through these points.    
<h3>Learn more    </h3>
  1. What are three collinear points on a line? brainly.com/question/5795008  
  2. Undefined terms are implemented to define a ray brainly.com/question/1087090
  3. Definition of the line segment brainly.com/question/909890    

Keywords: the undefined term, is needed to define, an angle, line, point, line, plane, ray, endpoint, acute, obtuse, right, straight, Euclidean geometry

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