Question
If secθ + tanθ = 5/2 for an acute angle θ, find
sinθ.Solution
Let S = secθ, T = tanθ. Given S + T = 5/2 Use identity: (secθ + tanθ)(secθ − tanθ) = 1 So (S − T) = 1/(S + T) = 1 / (5/2) = 2/5 Now: S + T = 5/2 S − T = 2/5 Add: 2S = 5/2 + 2/5 = (25/10 + 4/10) = 29/10 ⇒ S = 29/20 ⇒ secθ = 29/20 Thus cosθ = 1/secθ = 20/29 Subtract equations to find T: (S + T) − S = T = 5/2 − 29/20 = (50/20 − 29/20) = 21/20 ⇒ tanθ = 21/20 Now sinθ = tanθ × cosθ = (21/20) × (20/29) = 21/29 So sinθ = 21/29.
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