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

• 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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