Monday 6 May 2019

Trigonometry Questions and Answers

Trigonometry Questions and Answers



Solved Examples

Question 1: Prove the following identity:

Solution:
LHS = (secθ - coseθ)(1 + tanθ + cotθ)
= (sinθcosθsinθcosθ)(sinθcosθ+1sinθcosθ)
[Using identities, secθ = 1cosθ, coseθ = 1sinθ, tanθ = sinθcosθ and cotθ = cosθsinθ]
= (sinθcosθ)(sinθ cosθ+1)sin2θ cos2θ
= (sinθcosθ)(sinθ cosθ+sin2θ+cos2θ)sin2θ cos2θ
[Using, sin2θ+cos2θ = 1]





= tanθ sec θ - cotθcoscθ
= RHS

Question 2:Prove the following identity:


Solution:
LHS = 1 + 2sec2A tan2A - sec4A - tan4A

= 1 - (sec4A - 2sec2A tan2A + tan4A)





[Using identity, sec2A - tan2A = 1]

= 1 - 1

= 0

= RHS
Question 3: Prove the following identity:

Solution:
LHS = [1sinAcosAcosA(secAcosecA)][sin2Acos2Asin3A+cos3A]

= [1sinAcosAcosA(1cosA1sinA)][(sinA+cosA)(sinAcosA)(sinA+cosA)(sin2AsinAcosA+cos2A)]


[Using identities, secθ = 1cosθ, coseθ = 1sinθ, sin2A - cos2A = (sin A + cos A)(sin A - cos A) and sin3A + cos3A = (sin A + cos A)(sin2 A - sin A cos A + cos2A)]

[1sinAcosAsinAcosAsinA][sinAcosAsin2AsinAcosA+cos2A] [By cancelling common terms]

= sin A[1sinAcosAsinAcosA] * [sinAcosA1sinAcosA]

[Using identity sin2 A + cos2 A = 1]

= sin A (By cancelling common terms)

= RHS

Trigonometry Test Questions

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Lets us solve some more trigonometric examples using their identities:

Solved Examples

Question 1: Prove the following identity:


Solution:
LHS = (secθ - 1)2 - (tanθ - sinθ)2

= (1cosθ - 1)2 - (sinθcosθ - sinθ)2

= (1cosθcosθ)2 - sin2θcos2θ(1 - sinθ * cosθsinθ)2

= (1cosθcosθ)2 - (1 - cosθ)2 sin2θcos2θ
= (1 - cosθ)2 [1cos2θsin2θcos2θ]




Question 2:Prove the following identity:


Solution:
LHS = tan3θ1+tan2θ + cot3θ1+cot2θ

tan3θsec2θ + cot3θcose2θ











Trigonometry Sample Questions

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Solved Examples

Question 1:

Solution:
If tan A + sin A = m .................. (1)
tan A - sin A = n ................... (2)
Step 1:Adding (1) and (2)

Step 2:
Subtracting (2) from (1)

Now


[Using, (a + b)2 = a2 + b2 + 2ab and (a - b)2 = a2 + b2 - 2ab]



Question 2: 
Solution:








12+1 * 2121 cos θ

= (2 - 1)cos θ






Question 3:If x sin3θ + y cos3θ = sin θ cos θ and x sin θ - y cos θ = 0. Prove that x2 + y2 = 1.

Solution:
x sin3θ + y cos3θ = sin θ cos θ ..................... (i)

x sin θ - y cos θ = 0 ...................... (ii)



from (i) and (iii)







y = sinθ .................. (iv)

From (iii) and (iv), x = cosθ
=> x2 + y2 = sin2θ + cos2θ