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Question
If cotθ = `3/4` and π < θ < `(3pi)/2` then find the value of 4cosecθ + 5cosθ.
Solution
We know that,
cosec2θ = 1 + cot2θ
= `1 + (3/4)^2 = 1 + 9/16`
∴ cosec2θ = `25/16`
∴ cosecθ = `±5/4`
Since π < θ < `(3pi)/2`,
θ lies in the third quadrant.
∴ cosecθ < 0
∴ cosecθ = `-5/4`
cotθ = `3/4`
tanθ = `1/cottheta = 4/3`
We know that,
sec2θ = 1 + tan2θ = `1 + (4/3)^2`
= `1 + 16/9 = 25/9`
∴ secθ = `±5/3`
Since θ lies in the third quadrant,
secθ < 0
∴ secθ = `-5/3`
cosθ = `1/sectheta = (-3)/5`
∴ 4cosecθ + 5cosθ
= `4(-5/4) + 5(-3/5)`
= –5 – 3= – 8
Notes
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