Eliminate Theta between the 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 tan 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 sec θ

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

⇒ x2 – y2 = – tan2 θ (p2 – q2) + sec2 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 sec2 θ – tan2 θ = 1]

⇒ x2 – y2 = p2 – q

Hence the required eliminant is x2 – y = p2 – q2.

1. 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, [using (A)] ………………… (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.

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