Problems on Eliminate Theta

Here we will solve various types of problems on eliminate theta from the given equations.

We know, “eliminate theta from the equations” means that the equations are combined in such a way into one equation that it remains valid without the theta (θ) appearing in this new equation.


Worked-out problems on eliminate theta (θ) between the equations:

1. Eliminate theta between the equations:

x = a sin θ + b cos θ and y = a cos θ – b sin θ

                             OR,

If x = a sin θ + b cos θ and y = a cos θ –b sin θ, prove that

x2 + y2 = a2 + b2.



Solution:

We have x2 + y2 = (a sin θ + b cos θ)2 + (a cos θ – b sin θ)2

= (a2 sin2 θ + b2 cos2 θ + 2ab sin θ cos θ) + (a2 cos2 θ + b2 sin2 θ - 2ab sin θ cos θ)

= a2 sin2 θ + b2 cos2 θ + 2ab sin θ cos θ + a2 cos2 θ + b2 sin2 θ - 2ab sin θ cos θ

= a2 sin2 θ + b2 cos2 θ + a2 cos2 θ + b2 sin2 θ

= a2 sin2 θ + a2 cos2 θ + b2 sin2 θ + b2 cos2 θ

= a2 (sin2 θ + cos2 θ) + b2 (sin2 θ + cos2 θ)

= a2 (1) + b2 (1); [since, sin2 θ + cos2 θ = 1]

= a2 + b2

Therefore, x2 + y2 = a2 + b2

which is the required θ-eliminate.


2. Using the trig-identity we will solve the problems on eliminate theta (θ) between the equations:

tan θ - cot θ = a and cos θ + sin θ = b.

Solution:

tan θ – cot θ = a ………. (A)

cos θ + sin θ = b ………. (B)

Squaring both sides of (B) we get,

cos2 θ + sin2 θ + 2cos θ sin θ = b2

or, 1 + 2 cos θ sin θ = b2

or, 2 cos θ sin θ = b2 - 1 ………. (C)

Again, from (A) we get, (sin θ/cos θ) – (cos θ/sin θ) = a

or, (sin2 θ - cos2 θ)/(cos θ sin θ) = a

or, sin2θ - cos2θ = a sin θ cos θ

or, (sin θ + cos θ) (sin θ - cos θ) = a ∙ (b2 - 1)/2 ………. [by (C)]

or, b(sin θ - cos θ)= (½) a (b2 - 1) [by (B)]

or, b2 (sin θ - cos θ)2 = (1/4) a2 (b2 - 1)2, [Squaring both the sides]

or, b2 [(sin θ + cos θ)2 - 4 sinθ cos θ] = (1/4) a2 (b2 - 1)2

or, b2 [b2 - 2 ∙ (b2 - 1)] = (1/4) a2 (b2 - 1)2 [from (B) and (C)]

or, 4b2 (2 - b2) = a2 (b2 - 1)2

which is the required θ-eliminate.


Show how to use the trigonometric identities to solve the problems on eliminate theta form the given two equations.

3. x sin θ - y cos θ = √(x2 + y2) and cos2 θ/a2 + sin2 θ/b2 = 1/(x2 + y2)

Solution:

x sin θ - y cos θ = √(x2 + y2) ..........…. (A)

cos2 θ/a2 + sin2 θ/b2 = 1/(x2 + y2) ..........…. (B)

Squaring both sides of (A) we get,

x2 sin2 θ + y2 cos2 θ - 2xy sin θ cos θ = x2 + y2

or, x2 (1 - sin2 θ) + y2 (1 - cos2 θ) + 2xy sin θ cos θ = 0

or, x2 cos2 θ + y2 sin2 θ + 2 ∙ x cos θ ∙ y sin θ = 0

or, (x cos θ + y sin θ)2 = 0

or, x cos θ + y sin θ = 0

or, x cos θ = - y sin θ

or, cos θ/(-y) = sin θ/x

or, cos2 θ/y2 = sin2 θ/x2 = (cos2 θ + sin2 θ)/(y2 + x2) = 1/(x2 + y2)

Therefore, cos2 θ = y2/(x2 + y2) and sin2 θ = x2/(x2 + y2 )

Putting the values of cos2 θ and sin2 θ in (B) we get,

(1/a2) ∙ {y2/(x2} + y2) + (1/b2) ∙ {x2/(x2 + y2)} = 1/(x2 + y2)

Or, y2/a2 + x2/b2 = 1 (Since, x2 + y2 ≠0)

which is the required θ-eliminate.


The explanation will help us to understand how the steps are used technically to work-out the problems on eliminate theta form the given equations.

 Trigonometric Functions







10th Grade Math

From Problems on Eliminate Theta 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. Adding 5-digit Numbers with Regrouping | 5-digit Addition |Addition

    Mar 18, 24 02:31 PM

    Adding 5-digit Numbers with Regrouping
    We will learn adding 5-digit numbers with regrouping. We have learnt the addition of 4-digit numbers with regrouping and now in the same way we will do addition of 5-digit numbers with regrouping. We…

    Read More

  2. Adding 4-digit Numbers with Regrouping | 4-digit Addition |Addition

    Mar 18, 24 12:19 PM

    Adding 4-digit Numbers with Regrouping
    We will learn adding 4-digit numbers with regrouping. Addition of 4-digit numbers can be done in the same way as we do addition of smaller numbers. We first arrange the numbers one below the other in…

    Read More

  3. Worksheet on Adding 4-digit Numbers without Regrouping | Answers |Math

    Mar 16, 24 05:02 PM

    Missing Digits in Addition
    In worksheet on adding 4-digit numbers without regrouping we will solve the addition of 4-digit numbers without regrouping or without carrying, 4-digit vertical addition, arrange in columns and add an…

    Read More

  4. Adding 4-digit Numbers without Regrouping | 4-digit Addition |Addition

    Mar 15, 24 04:52 PM

    Adding 4-digit Numbers without Regrouping
    We will learn adding 4-digit numbers without regrouping. We first arrange the numbers one below the other in place value columns and then add the digits under each column as shown in the following exa…

    Read More

  5. Addition of Three 3-Digit Numbers | With and With out Regrouping |Math

    Mar 15, 24 04:33 PM

    Addition of Three 3-Digit Numbers Without Regrouping
    Without regrouping: Adding three 3-digit numbers is same as adding two 3-digit numbers.

    Read More