sin θ = sin ∝

How to find the general solution of an equation of the form sin θ = sin ∝?

Prove that the general solution of sin θ = sin ∝ is given by θ = nπ + (-1)\(^{n}\) ∝, n ∈ Z.

Solution:

We have,

sin θ = sin ∝            

⇒ sin θ - sin ∝ = 0 

⇒ 2 cos \(\frac{θ  +  ∝}{2}\) sin \(\frac{θ  -  ∝}{2}\) = 0

Therefore either cos \(\frac{θ  +  ∝}{2}\) = 0 or, sin \(\frac{θ  -  ∝}{2}\) = 0

Now, from cos \(\frac{θ  +  ∝}{2}\) = 0 we get, \(\frac{θ  +  ∝}{2}\) = (2m + 1)\(\frac{π}{2}\), m ∈ Z

⇒ θ = (2m + 1)π - ∝, m ∈ Z i.e., (any odd multiple of π) - ∝ ……………….(i)

And from sin \(\frac{θ  -  ∝}{2}\) = 0 we get,

\(\frac{θ  -  ∝}{2}\) = mπ, m ∈ Z                  

⇒ θ = 2mπ + ∝, m ∈ Z i.e., (any even multiple of π) + ∝ …………………….(ii)

Now combining the solutions (i) and (ii) we get,

θ = nπ + (-1)\(^{n}\) , where n ∈ Z.

Hence, the general solution of sin θ = sin ∝ is θ = nπ + (-1)\(^{n}\) , where n ∈ Z.

Note: The equation csc θ = csc ∝ is equivalent to sin θ = sin ∝ (since, csc θ = \(\frac{1}{sin  θ}\) and csc ∝ = \(\frac{1}{sin  ∝}\)). Thus, csc θ = csc ∝ and sin θ = sin ∝ have the same general solution.

Hence, the general solution of csc θ = csc ∝ is θ = nπ + (-1)\(^{n}\) , where n ∈ Z.


1. Find the general values of x which satisfy the equation sin 2x = -\(\frac{1}{2}\)

solution:

sin 2x = -\(\frac{1}{2}\)

sin 2x = - sin \(\frac{π}{6}\)

⇒ sin 2x = sin (π + \(\frac{π}{6}\))

⇒ sin 2x = sin \(\frac{7π}{6}\)

⇒ 2x = nπ + (-1)\(^{n}\) \(\frac{7π}{6}\), n ∈ Z

⇒ x = \(\frac{nπ}{2}\) + (-1)\(^{n}\) \(\frac{7π}{12}\), n ∈ Z

Therefore the general solution of sin 2x = -\(\frac{1}{2}\) is x = \(\frac{nπ}{2}\) + (-1)\(^{n}\) \(\frac{7π}{12}\), n ∈ Z


2. Find the general solution of the trigonometric equation sin 3θ = \(\frac{√3}{2}\).

Solution:

sin 3θ = \(\frac{√3}{2}\)

⇒ sin 3θ = sin \(\frac{π}{3}\)

⇒ 3θ = = nπ + (-1)\(^{n}\) \(\frac{π}{3}\), where, n = 0, ± 1, ± 2, ± 3, ± 4 .....

⇒ θ = \(\frac{nπ}{3}\) + (-1)\(^{n}\) \(\frac{π}{9}\),where, n = 0, ± 1, ± 2, ± 3, ± 4 .....

Therefore the general solution of sin 3θ = \(\frac{√3}{2}\) is θ = \(\frac{nπ}{3}\) + (-1)\(^{n}\) \(\frac{π}{9}\), where, n = 0, ± 1, ± 2, ± 3, ± 4 .....


3. Find the general solution of the equation csc θ = 2

Solution:

csc θ = 2

⇒ sin θ = \(\frac{1}{2}\)

⇒ sin θ = sin \(\frac{π}{6}\)

⇒ θ = nπ + (-1)\(^{n}\) \(\frac{π}{6}\), where, n ∈ Z, [Since, we know that the general solution of the equation sin θ = sin ∝ is θ = 2nπ + (-1)\(^{n}\) ∝, where n = 0, ± 1, ± 2, ± 3, ……. ]

Therefore the general solution of csc θ = 2 is θ = nπ + (-1)\(^{n}\) \(\frac{π}{6}\), where, n ∈ Z


4. Find the general solution of the trigonometric equation sin\(^{2}\) θ = \(\frac{3}{4}\).

Solution:

sin\(^{2}\) θ = \(\frac{3}{4}\).

sin θ = ± \(\frac{√3}{2}\)

sin θ = sin (± \(\frac{π}{3}\))

θ = nπ + (-1)\(^{n}\) ∙ (±\(\frac{π}{3}\)), where, n ∈ Z

θ = nπ ±\(\frac{π}{3}\), where, n ∈ Z

Therefore the general solution of sin\(^{2}\) θ = \(\frac{3}{4}\) is θ = nπ ±\(\frac{π}{3}\), where, n ∈ Z

 Trigonometric Equations





11 and 12 Grade Math

From sin θ = sin ∝ to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Dec 01, 23 01:16 AM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More

  2. Days of the Week | 7 Days of the Week | What are the Seven Days?

    Nov 30, 23 10:59 PM

    Days of the Weeks
    We know that, seven days of a week are Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. A day has 24 hours. There are 52 weeks in a year. Fill in the missing dates and answer the questi…

    Read More

  3. Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical

    Nov 30, 23 01:08 PM

    Types of Lines
    What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…

    Read More