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

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit of Trigonometric Ratios
  
• Trigonometrical Identity
• Problems on Trigonometric Identities
• Elimination of Trigonometric Ratios
• Eliminate Theta between the equations
• Problems on Eliminate Theta
• Trig Ratio Problems
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

10th Grade Math

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