If cos x = 4/5, 0° x 90°, find the value
of (1 + tan x)/(1 - tan x)
In QI, if cosx = 4/5, sinx = 3/5
(1+tanx)/(1-tanx) = (cosx+sinx)/(cosx-sinx)
= (4/5 + 3/5)/(4/5 - 3/5)
= (7/5)/(1/5)
= 7
cos x = 4 / 5
sin x = ±√ ( 1 - sin^2 x )
The angles which lie between 0° and 90° are lie in the first quadrant.
In the first quadrant sin x is positive so:
sin x = √ ( 1 - sin^2 x )
sin x = √ ( 1 - sin^2 x ) = sin x = √ [ 1 - ( 4 / 5 ) ^ 2 ] =
√ [ 1 - 16 / 25 ] = √ [ 25 / 25 - 16 / 25 ] =
√ [ 9 / 25 ] = √ 9 / √ 25 = 3 / 5
sin x = 3 / 5
Now:
tan x = sin x / cos x
tan x = ( 3 / 5 ) / ( 4 / 5 ) = 3 * 5 / 4 * 5 = 3 / 4
tan x = 3 / 4
( 1 + tan x ) / ( 1 - tan x ) =
( 1 + 3 / 4 ) / ( 1 - 3 / 4 ) =
( 4 / 4 + 3 / 4 ) / ( 4 / 4 - 3 / 4 ) =
( 7 / 4 ) / ( 1 / 4 ) = 7 * 4 / 1 * 4 = 7 / 1 = 7
To find the value of (1 + tan x)/(1 - tan x), where cos x = 4/5 and 0° ≤ x ≤ 90°, we need to find the value of tan x.
Step 1: Use the Pythagorean identity to find the value of sin x:
sin x = √(1 - cos²x)
= √(1 - (4/5)²)
= √(1 - 16/25)
= √(9/25)
= 3/5
Step 2: Use the tangent identity to find the value of tan x:
tan x = sin x / cos x
= (3/5) / (4/5)
= 3/4
Step 3: Substitute the value of tan x into (1 + tan x)/(1 - tan x):
(1 + tan x)/(1 - tan x) = (1 + 3/4) / (1 - 3/4)
= (4/4 + 3/4) / (4/4 - 3/4)
= (7/4) / (1/4)
= 7/1
= 7
Therefore, the value of (1 + tan x)/(1 - tan x) is 7.
To find the value of (1 + tan x)/(1 - tan x) given that cos x = 4/5, we can use the trigonometric identity involving tangent:
tan(x) = sin(x)/cos(x)
Let's first find the value of sin(x) using the Pythagorean identity:
sin(x) = √(1 - cos^2(x))
Substituting the given value of cos(x) = 4/5:
sin(x) = √(1 - (4/5)^2)
= √(1 - 16/25)
= √(25/25 - 16/25)
= √(9/25)
= 3/5
Now, substitute the values of sin(x) and cos(x) into the expression (1 + tan x)/(1 - tan x) to get the final answer:
(1 + tan x)/(1 - tan x) = (1 + sin(x)/cos(x))/(1 - sin(x)/cos(x))
= [(cos(x) + sin(x))/cos(x)] / [(cos(x) - sin(x))/cos(x)]
= (cos(x) + sin(x)) / (cos(x) - sin(x))
= [(4/5) + (3/5)] / [(4/5) - (3/5)]
= (7/5) / (1/5)
= 7
Therefore, the value of (1 + tan x)/(1 - tan x) is 7.