Example
If asinѳ + bcosѳ = p
andacosѳ – bsinѳ = q
show that a2 + b2 = p2 + q2
Solution
asinѳ + bcosѳ = p …(1)
acosѳ – bsinѳ = q …(2)
Squaring both sides of (1)
(asinѳ + bcos)2 = p2
: a2 sin2ѳ + 2absinѳ cosѳ + b2cos2ѳ = p2 …(3)
Squaring both sides of (2)
(acosѳ – bsinѳ)2 = q2
a2cos2ѳ – 2absinѳ cosѳ + b2cos2ѳ = q2 …(4)
Adding (3) and (4)
a2sin2ѳ + 2absinѳ cosѳ + b2cos2ѳ + a2cosѳ – 2absinѳ cosѳ + b2sin2ѳ = p2q2
: a2sin2ѳ + a2cos2ѳ + b2cos2ѳ + b2sin2ѳ = p2 + q2
a2(sin2ѳ + cos2ѳ) + b2(cos2ѳ + sin2ѳ) = p2 + q2
But sin2ѳ + cos2ѳ = 1
: a2 + b2 = p2 + q2
Evaluation
Sketch the graph of:
(i) y = sin2x (ii) y = cos x
(iii) y = sec x (iv) cosec x
all at intervals of 30◦ range 0≤ x ≤ 360.
General Evaluation
(1) Draw the graph of y = 2cosx – 1 in the range 0◦ ≤ x ≤ 360◦ at intervals of 30◦.
(2) Draw the graph of y = 3sin x – 1 in the range of 0◦ ≤ x ≤ 360◦ at intervals of 30◦
(3) Prove that sec2ѳ + cosec2ѳ = (tanѳ + cotѳ) 2.
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