Subscribe to our YouTube channel for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
Problems on elimination of unknown angles using trigonometric identities.
1. If x = tan θ + sin θ and y = tan θ - sin θ, prove that x2 – y2 = 4√xy.
Solution:
Given that
x = tan θ + sin θ ……………………. (i)
and
y = tan θ - sin θ ……………………. (ii)
Adding (i) and (ii), we get
x + y = 2 tan θ ……………………. (iii)
⟹ tan θ = x+y2 ……………………. (iv)
Subtracting (ii) from (i), we get,
x - y = 2 sin θ ……………………. (v)
Now, dividing (iii) by (v) we get,
x+yx−y = 2tanθ2sinθ
= tanθsinθ
= sinθcosθsinθ
= sinθcosθ ∙ 1sinθ
= 1cosθ
= sec θ
Therefore, sec θ = x+yx−y ……………………. (vi)
We know that the Pythagorean identity, sec2 θ - tan2 θ = 1.
Now from (iv) and (vi) we get,
(x+yx−y)2 - (x+y2)2 = 1
Taking common (x + y)2 we get,
⟹ (x + y)2 ∙ {1(x−y)2−14} = 1
⟹ (x + y)2 ∙ 4–(x–y)24(x–y)2= 1
⟹ (x + y)2 ∙ {4 – (x – y)2} = 4(x – y)2
⟹ 4(x + y)2 - (x + y)2 ∙ (x – y)2 = 4(x – y)2
⟹ 4(x + y)2 - 4(x – y)2 = (x + y)2 ∙ (x – y)2
⟹ 4(x2 + y2 + 2xy - x2 - y2 + 2xy) = (x2+y2)2
⟹ 4 ∙ 4xy = (x2+y2)2
⟹ 16xy = (x2+y2)2
⟹ 4√xy = x2+y2
Therefore, x2+y2 = 4√xy. (Proved)
2. If a = r cos θ ∙ sin β, b = r cos θ ∙ cos β and c = r sin θ then prove that a2 + b2 + c2 = r2.
Solution:
a2 + b2 + c2 = r2 cos2 θ ∙ sin2 β + r2 cos2 θ ∙ cos2 β + r2 sin2 θ
= r2 cos2 θ(sin2 β + cos2 β) + r2 sin2 θ
= r2 cos2 θ ∙ (1) + r2 sin2 θ, [since We know that the Pythagorean identity, sin2 θ + cos2 θ = 1.]
= r2 cos2 θ + r2 sin2 θ
= r2 (cos2 θ + sin2 θ)
= r2 ∙ (1), [since, sin2 θ + cos2 θ = 1]
= r2
Therefore, a2 + b2 + c2 = r2. (proved)
From Elimination of Unknown Angles 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.
Jul 11, 25 02:14 PM
Jul 09, 25 01:29 AM
Jul 08, 25 02:32 PM
Jul 08, 25 02:23 PM
Jul 08, 25 09:55 AM
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.