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. Intersecting Lines | What Are Intersecting Lines? | Definition

    Jun 14, 24 11:00 AM

    Intersecting Lines
    Two lines that cross each other at a particular point are called intersecting lines. The point where two lines cross is called the point of intersection. In the given figure AB and CD intersect each o…

    Read More

  2. Line-Segment, Ray and Line | Definition of in Line-segment | Symbol

    Jun 14, 24 10:41 AM

    Line-Segment, Ray and Line
    Definition of in Line-segment, ray and line geometry: A line segment is a fixed part of a line. It has two end points. It is named by the end points. In the figure given below end points are A and B…

    Read More

  3. Definition of Points, Lines and Shapes in Geometry | Types & Examples

    Jun 14, 24 09:45 AM

    How Many Points are There?
    Definition of points, lines and shapes in geometry: Point: A point is the fundamental element of geometry. If we put the tip of a pencil on a paper and press it lightly,

    Read More

  4. Subtracting Integers | Subtraction of Integers |Fundamental Operations

    Jun 13, 24 04:32 PM

    Subtraction of Integers
    Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

    Read More

  5. 6th Grade Worksheet on Whole Numbers |Answer|6th Grade Math Worksheets

    Jun 13, 24 04:17 PM

    6th Grade Worksheet on Whole Numbers
    In 6th Grade Worksheet on Whole Numbers contains various types of questions on whole numbers, successor and predecessor of a number, number line, addition of whole numbers, subtraction of whole number…

    Read More