sin θ = 1

How to find the general solution of an equation of the form sin θ = 1?

Prove that the general solution of sin θ = 1 is given by θ = (4n + 1)π/2, n ∈ Z.

Solution:

We have,

sin θ = 1       

⇒ sin θ = sin \(\frac{π}{2}\)

θ = mπ + (-1)\(^{m}\) ∙ \(\frac{π}{2}\), m ∈ Z, [Since, the general solution of sin θ = sin ∝ is given by θ = nπ + (-1)\(^{n}\) ∝, n ∈ Z.]

Now, if m is an even integer i.e., m = 2n (where n ∈ Z) then,

    θ = 2nπ + \(\frac{π}{2}\)

⇒ θ = (4n + 1)\(\frac{π}{2}\)

Again, if m is an odd integer i.e. m = 2n + 1 (where n ∈ Z) then,

θ = (2n + 1) ∙ π - \(\frac{π}{2}\)

⇒ θ = (4n + 1)\(\frac{π}{2}\).

Hence, the general solution of sin θ = 1 is θ = (4n + 1)\(\frac{π}{2}\), n ∈ Z.


1. Solve the trigonometric equation sin x - 2 = cos 2x, (0 ≤ x ≤ \(\frac{π}{2}\))

Solution:

sin x - 2 = cos 2x

⇒ sin x - 2 = 1 - 2 sin 2x

⇒ 2 sin\(^{2}\) x + sin x - 3 = 0

⇒ 2 sin\(^{2}\) x + 3 sin x - 2 sin x - 3 = 0

⇒ sin x (2 sin x + 3) - 1(2 sin x + 3) = 0

⇒ (2 sin x + 3) (sin x - 1) = 0

Therefore, either, 2 sin x + 3 = 0 ⇒ sin x = - \(\frac{3}{2}\), Which is impossible since the numerical value of sin x cannot be greater than 1.

or, sin x - 1 = 0 

⇒ sin x = 1

We know that the general solution of sin θ = 1 is θ = (4n + 1)\(\frac{π}{2}\), n ∈ Z.

Therefore, x = (4n + 1)\(\frac{π}{2}\) …………… (1) where, n ∈ Z.

Now, Putting n = 0 in (1) we get, x = \(\frac{π}{2}\)

Now, Putting   n = 1 in (1) we get, x = \(\frac{5π}{2}\)

Therefore, the required solution in 0 ≤ x ≤ 2π is:   x = \(\frac{π}{2}\).

 Trigonometric Equations






11 and 12 Grade Math

From sin θ = 1 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. Measuring Capacity | Standard Unit of Capacity | Litre | Millilitres

    Nov 29, 23 01:15 AM

    2 Tablespoonful of Water
    We will discuss about measuring capacity. The milkman measures milk in liters. Petrol is given in liters. Mobil oil is sold in liters. Two milk bottles contain 1 liter of milk. One milk bottle

    Read More

  2. Addition and Subtraction of Units of Measurement | Metric Units

    Nov 29, 23 12:54 AM

    Addition of Lengths
    We can add the units of measurement like decimal numbers. 1. Add 5 m 9 dm and 11 m and 5 dm Solution: 5 m 9 dm = 5.9 m 11 m 5 dm = 11.5 m Hence, 5 m 9 dm + 11 m 5 dm = 17 m 4 dm or 17.4 m 2. Add 15 cm…

    Read More

  3. 1 to 10 Times Tables | 1 - 10 Times Table Chart |Multiplication Tables

    Nov 29, 23 12:50 AM

    1 to 10 Times Tables
    Memorizing 1 to 10 Times Tables are very important for mental math and quick calculations. Times Tables are used during multiplication and division. Let us learn all the times tables from 1 to 10 to i…

    Read More