Eliminate Theta Between The Equations

Here we will learn how to eliminate theta between the equations with the help of different types of problems.

Before we solve different types of question on eliminating θ (theta) let us understand what does it mean to "Eliminate theta from two or more equations"?

“Eliminate theta from two or more equations” means that the two or more equations are combined logically into one equation that it remains valid and theta (θ) does not appear in this new equation.


Worked-out examples to eliminate theta between the equations:

1. Eliminate theta between the equations:

q tan θ + p sec θ = x, p tan θ + q sec θ = y



Solution:

Squaring both sides of q tan θ + p sec θ = x we get,

(q tan θ + p sec θ)2 = x2 , …………….. (A)

Now, squaring both sides of p tan θ + q sec θ = y we get,

(p tan θ + q sec θ)2 = y2, …………….. (B)

Now subtract (B) from (A) we get,

x2 - y2 = (q tan θ + p sec θ)2 - (p tan θ + q sec θ) 2

⇒ x2 - y2 = (q2 tan2 θ + p2 sec2 θ + 2qp tan θ sec θ) - (p2 tan2 θ + q2 sec2 θ + 2pq tan θ sec θ)

⇒ x2 - y2 = q2 tan2 θ + p2 sec2 θ + 2qp tan θ sec θ - p2 tan2 θ - q2 sec2 θ - 2pq tan θ sec θ

⇒ x2 - y2 = q2 tan2 θ - p2 tan2 θ + p2 sec2 θ - q2 sec2 θ

⇒ x2 - y2 = tan2 θ (q2 – p2) + sec 2 θ (p2 - q2)

⇒ x2 - y2 = - tan2 θ (p2 - q2) + sec 2 θ (p2 - q2) ⇒ x2 - y2 = sec2 θ (p2 - q2) - tan2 θ (p2 - q2)

⇒ x2 - y2 = (p2 – q2) (sec2 θ - tan2 θ)

⇒ x2 - y2 = (p2 – q2)(1), [Since sec 2 θ - tan2 θ = 1]

⇒ x2 - y2 = p2 – q2

Hence the required eliminant is x2 - y2 = p2 - q2.


2. Eliminate theta between the equations: cos θ + sin θ = m and sec θ + csc θ = n.

                                       OR,

If cos θ + sin θ = m and sec θ + csc θ = n, prove that n(m2 – 1) = 2m.

Solution:

Given sec θ + csc θ = n ………………… (A)

⇒ 1/cos θ + 1/sin θ = n

⇒ (sin θ + cos θ)/cos θ sin θ = n

cos θ sin θ = (sin θ + cos θ)/n

sin θ = m/n, [Given, cos θ + sin θ = m] ………………… (B)

Now cos θ + sin θ = m

⇒ (cos θ + sin θ)2 = m2

⇒ cos2 θ + sin2 θ + 2 sin θ cos θ = m2

⇒ 1 + 2 sin θ cos θ = m2

⇒ 1 + 2 ∙ (m/n) = m2, [Using (B)]

⇒ 2 (m/n) = m2 - 1

⇒ 2m = n(m2 - 1), [Proved]


The above problems to eliminate theta between the equations are explained step-by-step so, that students get the clear concept how the equations are logically combined and it remains valid without the theta (θ) in the new equation.

Trigonometric Functions



10th Grade Math

From Eliminate Theta Between the Equations to HOME PAGE




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.



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.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Comparing and Ordering Decimals |Arranging Decimals

    Apr 19, 25 12:16 PM

    Arranging Decimals
    Practice different types of math questions given in the worksheet on comparing and ordering decimals. This worksheet contains questions mainly related to compare decimals and then place the decimals i…

    Read More

  2. Comparison of Decimal Fractions | Comparing Decimals Numbers | Decimal

    Apr 19, 25 11:47 AM

    Comparison of Decimal Fractions
    While comparing natural numbers we first compare total number of digits in both the numbers and if they are equal then we compare the digit at the extreme left. If they also equal then we compare the…

    Read More

  3. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    Apr 19, 25 11:25 AM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  4. Missing Numbers up to 10 | Worksheets on Missing Numbers up to 10

    Apr 18, 25 04:53 PM

    missing numbers up to 10
    Printable worksheets on missing numbers up to 10 help the kids to practice counting of the numbers.

    Read More

  5. Ordering Decimals | Comparing Decimals | Ascending & Descending Order

    Apr 18, 25 01:49 PM

    Ordering Decimal Numbers
    In ordering decimals we will learn how to compare two or more decimals. (i) Convert each of them as like decimals. (ii) Compare these decimals just as we compare two whole numbers ignoring

    Read More