Question 1 :
If cos θ : sin θ = 1 : 2, then find the value of (8 cos θ - 2 sin θ)/(4 cos θ + 2 sin θ)
Solution :
Given that :
cos θ : sin θ = 1 : 2
cos θ/sin θ = 1/2
cot θ = 1/2 = Adjacent side / Opposite side
(Hypotenuse side)2 = (Opposite side)2 + (Adjacent side)2
= 12 + 22
(Hypotenuse side)2 = 5
Hypotenuse side = √5
sin θ = 1/√5, cos θ = 2/√5
(8 cos θ - 2 sin θ)/(4 cos θ + 2 sin θ)
= (8/√5) - (4/√5)/(4/√5) + 2(2/√5)
= (4/√5)/(8/√5)
= 4/8
= 1/2
Question 2 :
From the given figure, prove that θ + Φ = 90°.Also prove that there are two other right angled triangles. Find sin a, cos β and tan Φ.
Solution :
In triangle ACD,
sin a = Opposite side / Hypotenuse side
sin a = DC/AC
sin a = 12/15 = 4/5
In triangle BDC,
cos β = Adjacent side / Hypotenuse side
= DB/BC
cos β = 16/20 = 4/5
tan Φ = Opposite side / Adjacent side
tan Φ = DB/DC = 16/12
= 4/3
Question 3 :
A boy standing at a point O finds his kite flying at a point P with distance OP = 25 m. It is at a height of 5 m from the ground. When the thread is extended by 10 m from P, it reaches a point Q. What will be the height QN of the kite from the ground? (use trigonometric ratios)
Solution :
In triangle OPM,
tan θ = PM/OP
= 5/25 ---(1)
tan θ = ON / (OP + PQ)
= h/(25 + 10)
= h/35 ---(2)
(1) = (2)
5/25 = h/35
(1/5)(35) = h
h = 7
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