Given sinx= -4/5 and x is in quadrant 3, what is the value of tan x/2.

Solution:

Given the value of sinx = -4/5

x is in the third quadrant

We need to find the value of tan x/2

Writing sin x in terms of tan x/2 using the trigonometric formula,

sin x = (2 tan(x/2)) / (1 + tan²(x/2))

-4/5 = (2 tan(x/2)) / (1 + tan²(x/2))

if we replace tan (x/2) by y, we get a quadratic equation,

-4/5 = 2y/(1+y²)

-4 -4y² = 10y

2y² + 5y + 2 = 0

By using the quadratic formula, we get y = -0.5, -2

Hence, the value of tan(x/2) = -0.5, -2

We have two solutions of tan(x/2).

checking for the ideal solution using the formula,we get tan x = (2 tan(x/2)) / (1 - tan²(x/2)).

For tan(x/2) = -0.5:

tan x = 2(-0.5) / 1 - (-0.5)² = -4/3

x is in quadrant 3.

tan is positive in the third quadrant, and we get tan x = -4/3 which is negative.

we can say that tan(x/2) = -0.5 is not a correct solution. Hence it is rejected.

Checking for tan (x/2) = -2. We get,

tan x = 2(-2) / 1 - (-2)² = 4/3

we get tan x = 4/3 which is positive.

Therefore, the value of tan x/2 is -2.

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Given sinx = -4/5 and x is in quadrant 3, what is the value of tan x/2.

Summary:

Given sinx = -4/5 and x is in quadrant 3, the value of tan x/2 is -2.