Trigonometric Equations

4,246 views

Published on

0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
4,246
On SlideShare
0
From Embeds
0
Number of Embeds
7
Actions
Shares
0
Downloads
0
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

Trigonometric Equations

  1. 1. TRIGONOMETRIC EQUATIONS
  2. 2. TRIGONOMETRIC EQUATION An equation involving one or more trigonometric ratios of an unknown angle is called a trigonometric equation.
  3. 3. SOLUTION OF TRIGONOMETRIC EQUATION A solution of trigonometric equation is the value of the unknown angle that satisfies the equation. 3 9 11 e.g. if sin θ = 1/√2 θ= , , , ,...... 4 4 4 4 Thus, the trigonometric equation may have infinite number of solutions (because of their periodic nature) and can be classified as :i. Principal solutionii. General solution.
  4. 4.  Principal solutions: The solutions of a trigonometric equation which lie in the interval [0, 2 ) are called Principal solutions. General Solution : The expression involving an integer n which gives all solutions of a trigonometric equation is called General solution. General solution of some standard trigonometric equations are given in next slides.
  5. 5. GENERAL SOLUTION OF SOME STANDARDTRIGONOMETRIC EQUATIONS If sin θ = sin θ = n + (-1)n where a , 2 2 , n I. If cos θ = cos θ = 2n ± where a [0, ], n I. If tan θ = tan θ=n + where a , , 2 2 n I. If sin² θ = sin² θ=n ± n I. If cos² θ = cos² θ=n ± n I. If tan² θ = tan² θ=n ± n I. [ Note: is called the principal angle ]
  6. 6. SOME IMPORTANT DEDUCTIONS sinθ = 0 θ=n , n I sin θ = 1 θ = (4n + 1) , n I 2 sin θ = – 1 θ = (4n – 1) , n I 2 cos θ = 0 θ = (2n + 1) , n I 2 cos θ = 1 θ = 2n , n I cos θ = – 1 θ = (2n + 1) , n I tan θ = 0 θ=n , n I
  7. 7. EXAMPLES
  8. 8. EXAMPLES
  9. 9. TYPES OF TRIGONOMETRIC EQUATIONS Type -1 : Trigonometric equations which can be solved by use of factorization. Type - 2 : Trigonometric equations which can be solved by reducing them in quadratic equations. Type – 3 : Trigonometric equations which can be solved by transforming a sum or difference of trigonometric ratios into their product. Type – 4 : Trigonometric equations which can be solved by transforming a product of trigonometric ratios into their sum or difference. Type – 5 : Trigonometric Equations of the form a sin x + b cos x = c, where a, b, c R, can be solved by dividing both sides of the equation by a 2 b. 2
  10. 10. EXAMPLES
  11. 11. EXAMPLES
  12. 12.  Type – 6 : Trigonometric equations of the form P(sin x ± cos x, sin x cos x) = 0, where p(y, z) is a polynomial, can be solved by using the substitution sin x ± cos x = t. Type – 7 : Trigonometric equations which can be solved by the use of boundness of the trigonometric ratios sin x and cos x.
  13. 13. EXAMPLES

×