answer this question
Answers
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)