Question
Download Solution PDFIf tan θ + sec θ= 7, θ being acute, then the value of 5 sin θ is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
tanθ + secθ= 7
Concept used:
1. sec2θ - tan2θ = 1
2. tanα = sinα/cosα
3. secα = 1/cosα
Calculation:
tanθ + secθ = 7 ....(1)
We know,
sec2θ - tan2θ = 1
⇒ (secθ - tanθ)(secθ + tanθ) = 1
⇒ (secθ - tanθ) × 7 = 1 (From 1)
⇒ (secθ - tanθ) = 1/7 ....(2)
By adding (1) and (2) we get,
⇒ 2 secθ = 7 + 1/7 = 50/7
⇒ secθ = 25/7
By subtracting (1) and (2) we get,
⇒ 2 tanθ = 7 - 1/7 = 48/7
⇒ tanθ = 24/7
secθ = 25/7 and tanθ = 24/7
So, sinθ
⇒ tanθ ÷ secθ
⇒ 24/7 ÷ 25/7
⇒ 24/25
Now, 5sinθ
⇒ 5sinθ = 5 × 24/25 = 24/5
∴ The value of 5sinθ is \({{24} \over 5}\).
Shortcut Trick
sec θ + tan θ = 7 … (i)
Then, sec θ - tan θ = `1/7… (ii) (as, sec2θ - tan2θ = 1)
Adding the equations we get,
⇒ 2sec θ = 50/7
⇒ sec θ = 25/7 = Hypo/Base
So, sin θ = 24/25
⇒ 5sin θ = 24/25 × 5 = 24/5
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