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Answered by MisterIncredible
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Question

If 5 tan A = 4, find the value of (8sin A - 3sin A)/(8sin A + 2 cos A)

ANSWER

Given : -

If 5 tan A = 4

Required to find : -

  • Value of (8sin A - 3sin A)/(8sin A + 2 cos A)

Identities used : -

  • sin² A + cos² A = 1
  • sec² A - tan² A = 1

Solution : -

If 5 tan A = 4

tan A = (4)/(5)

Now,

Let's find sin A in terms of tan A

so,

we know that;

(sin A)/(cos A) = tan A

sin A = tan A*cos A

sin A = (tan A)/(sec A)

From the identity;

sec² A + tan² A = 1

sec A = √(1+tan² A)

This implies;

sin A = [tan A]/[√(1+tan² A)]

sin A = [(4)/(5)]/[√(1+{(4)/(5)}²)]

sin A = [(4)/(5)]/[√(1+(16)/(25))]

sin A = [(4)/(5)]/[√({25+16}/{25})]

sin A = [(4)/(5)]/[√({41})/{5}]

  • sin A = [4]/[√(41)]

similarly,

we know that;

(sin A)/(cos A) = tan A

cos A = tan A*sin A

cos A = (tan A)/(cosec A)

From the identity;

cosec² A - cot² A = 1

cosec² A = 1 + cot² A

cosec² A = 1 + (1)/(tan² A)

cosec² A = (tan² A + 1)/(tan² A)

cosec A = √[(tan² A + 1)/(tan² A)

cosec A = [√(tan² A +1)]/[tan A]

This implies;

cos A = (tan A)/([(√{tan² A + 1})/(tan A)])

cos A = ([4]/[5])/([(√{[{4}/{5}]² + 1})/({[4]/[5]})

cos A = ([4]/[5])/([(√{[16]/[25]+1})/([4]/[5])

cos A = ([4]/[5])/(√{[16+25]/[25])/([4]/[5])

cos A = ([4]/[5]) x ([5]/[4])/([√(41)]/[5])

cos A = ([4]/[5]) x ([5]/[4]) x ([5]/[√(41)])

  • cos A = (5)/(√[41])

Now,

Let's find the value of 8sin A - 3cos A

8sin A - 3cos A =

8 x [4]/[√(41)] - 3 x (5)/(√[41])

[36]/[√(41)] - (15)/(√[41])

[36 - 15]/[√(41)]

[21]/[√(41)]

similarly,

Value of 8sin A + 2cos A is

8sin A + 2cos A =

8 x [4]/[√(41)] + 2 x [5]/[√(41)]

[32]/[√(41)] + [10]/[√(41)]

[32 + 10]/[√(41)]

[42]/[√(41)]

Now,

Value of (8sin A - 3sin A)/(8sin A + 2 cos A) is;

( [21]/[√{41}] ) ÷ ( [42]/[√{41}] )

[21]/[√{41}] x [√(41)]/[42]

[1]/[2]

Therefore,

Value of (8sin A - 3sin A)/(8sin A + 2 cos A) is (1)/(2)

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