1.

If cos x = -3/5 and π < x < 3π/2 find the values of other five trigonometric functions and hence evaluate (cosec x + cot x)/(sec x - tan x)

Answer»

Given as

cos x = -3/5 and π < x < 3π/2

As we know that in the third quadrant, tan x and cot x are positive and all other rations are negative.

On using the formulas,

Sin x = – √(1 - cos2 x)

Tan x = sin x/cos x

Cot x = 1/tan x

Sec x = 1/cos x

Cosec x = 1/sin x

Then,

Sin x = – √(1 - cos2 x)

= – √(1 - (-3/5)2)

= – √(1 - 9/25)

= – √((25 - 9)/25)

= – √(16/25)

= – 4/5

Tan x = sin x/cos x

= (-4/5)/(-3/5)

= -4/5 × -5/3

= 4/3

Cot x = 1/tan x

= 1/(4/3)

= 3/4

Sec x = 1/cos x

= 1/(-3/5)

= -5/3

Cosec x = 1/sin x

= 1/(-4/5)

= -5/4

∴ (cosec x + cot x)/(sec x - tan x)
= [(-5/4) + (3/4)]/[(-5/3) – (4/3)]

= [(-5 + 3)/4]/[(-5 - 4)/3]

= [-2/4]/[-9/3]

= [-1/2]/[-3]

= 1/6



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