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Trigonometry 4 (Solving Trigonometric Equations ) - Using basic equations and trigonometric identities and properties to solve trigonometric equations



                                                                                                                        
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                                           Trigonometric Equations


Trigonometric Equations

 
Here's a very quick outline of what's been introduced in this tutorial :
  • Trigonometric Equations are the equations that require to solve for the value (or values) of the angle for which the given trigonometric expressions holds the particular value. The general form is: Expression = value, where expression consists of (most generally) trigonometric ratios.

Example: Consider the equation :

2sinx=30.5

We solve it to get:

sin x=30.5/2

Or, x=π/3,2π/3,4π/3,…


  • The fact that x can have a lot of values leads to the concept of a general solution and particular solutions. When x is constrained to lie in a certain range, then we get particular solutions, if not, then we get general solutions.

  • The basic equations that help in solving difficult trigonometric equations are :


sin x = sin y


cos x =cos y


tan x = tan y


For solving these equations, we need to be familiar with the solutions of equations:


sin x=0, cos x=0, cot x=0.


  • Solution of sin x = 0
    From the study of trigonometric functions, we know that sin x = 0 when x=0,π,2π,-π,-2π and so on. Thus the solution is x=nπ, where nεZ (the set of Integers)
    Solution of cos x=0.
    From the study of trigonometric equations, we know that cos x becomes 0 when x=π/2,3π/2,5π/2,-π/2,-3π/2,-5π/2,…
    Thus, the solution is x=(2n+1)π/2
    Solution of tan x=0 is the same as that of sin x=0 (because there is no such value of x for which both sine and cosine become 0 simultaneously).

  • Solution of sin x = sin y.

sin x=sin y
=>sin x-sin y=0
=>2sin(x-y)/2 cos(x+y)/2=0
=>sin (x-y)/2=0 or cos(x+y)/2=0
=>x-y=2nπ or x+y=(2m+1)π, m, nεZ
=>x=2nπ+y or x=(2m+1)π
=>x=(even multiple of π)+y or (odd multiple of π)-y
=>x=nπ+(-1)ny, nεZ
The solutions of cosec x= cosec y are also the same.


  • Solution of cos x= cos y

             cos x= cos y
Following the similar procedure, we get
-2sin(x+y)/2 sin (x-y)/2=0.
=>x+y=2nπ or x-y=2mπ
=>x=2nπ±y, nεZ
sec x=sec y has the same solution.


  • Solution of tan x=tan y.

tan x=tan y
=>sin x cos y=sinycosx
=>sin(x-y)=0
=>x-y=nπ
=>x=nπ+y , nεZ
cot x=cot y has the same solution


  • Solutions of sin2x=sin2y,cos2x=cos2y, tan2x=tan2y are x=nπ±y

  • This can be obtained by using trigonometric identities, and using one of the basic equations. One of these is solved in the questions.
  • While solving, avoid squaring, and if you do square both sides of equations, check that roots obtained actually satisfy the original equation.

Here are the kind of questions which have been solved in this tutorial :


Solve sec x = 0.25.
Solve sin5x=0; x is in the range of 0 to π
Solve sin x+ sin 3x+sin 5x=0
Solve sin nx +cos mx=0.
Solve sin3x+cos2x=0Solve cot2x+3/sinx+3=0
Solve sin2x=sin2y, and tan2x=tan2y
Solve √3cosx+sinx=√2
Solve for general x, y sin (x – y) = 2 sin x sin y, where x and y are two acute angles of right angle triangle.
Solve the equation sin4 x + cos4 x = 7/2 sinx cosx

Complete Tutorial (To test how much you know, try out our Quiz at the end):



MCQ Quiz #1

Companion MCQ Quiz for Trigonometric Equations (Basic Problems) - test how much you know about the topic ! Your score will be e-mailed to you at the address you provide.

MCQ Quiz: Basic Trigonometric Equations


MCQ Quiz #2

Companion MCQ Quiz for Trigonometric Equations (More Challenging Problems) - test how much you know about the topic ! Your score will be e-mailed to you at the address you provide.

Google Spreadsheet Form


 


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