Question
Download Solution PDFWhat is the maximum value of 7cos2θ + 5sin2θ
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
Let f(θ) = 7cos2θ + 5sin2θ
= 2cos2θ + 5cos2θ + 5sin2θ
= 2cos2θ + 5(cos2θ + sin2θ)
= 2cos2θ + 5 [∵ cos2θ + sin2θ = 1]
As we know, 0 ≤ cos2θ ≤ 1
⇒ 0 ≤ 2cos2θ ≤ 2
⇒ 0 + 5 ≤ 2cos2θ + 5 ≤ 2 + 5
⇒ 5 ≤ 2cos2θ + 5 ≤ 7
⇒ 5 ≤ f(θ) ≤ 7
So, Maximum value of function is 7.
Alternate Method
Concept:
If acos2θ + bsin2θ where, a > b
a will be the maximum value and b will be the minimum value
Calculation:
7cos2θ + 5sin2θ
According to concept
acos2θ + bsin2θ where, a > b
On comparing
⇒ a = 7, b = 5 where, 7 > 5
∴ maximum value of 7cos2θ + 5sin2θ is 7
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