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प्रश्न
If 5 sec θ – 12 cosec θ = 0, then find values of sin θ, sec θ
उत्तर
5 sec θ – 12 cosec θ = 0 ......[Given]
∴ 5 sec θ = 12 cosec θ
∴ `5/costheta = 12/sintheta` ......`[because sectheta = 1/costheta, "cosec" theta = 1/sintheta]`
∴ `sintheta/costheta = 12/5`
∴ tan θ = `12/5`
We know that,
1 + tan2θ = sec2θ
∴ `1 + (12/5)^2` = sec2θ
∴ `1 + 144/25` = sec2θ
∴ `(25 + 144)/25` = sec2θ
∴ sec2θ = `169/25`
∴ secθ = `13/5` ......[Taking square root of both sides]
Now, cos θ = `1/sectheta`
= `1/((13/5))`
∴ cos θ = `5/13`
We know that,
sin2θ + cos2θ = 1
∴ `sin^2theta + (5/13)^2` = 1
∴ `sin^2theta + 25/169` = 1
∴ sec2θ = `1 - 25/169`
∴ sec2θ = `(169 - 25)/169`
∴ sec2θ = `144/169`
∴ sin θ = `12/13` ......[Taking square root of both sides]
∴ sin θ = `12/13`, sec θ = `13/5`.
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