Question

Find the values of the other five trigonometric functions in each of the following:

(i) cot x = 12/5, x in quadrant III

(ii) cos x = -1/2, x in quadrant II

(iii) tan x = 3/4, x in quadrant III

(iv) sin x = 3/5, x in quadrant I

Answer 1

(i) Since, in third quadrant, tan x and cot x are positive. sin x, cos x, sec x, cosec x are negative.

On using the formulas,

tan x = 1/cot x

= 1/(12/5)

= 5/12

cosec x = –√(1 + cot² x)

= –√(1 + (12/5)²)

= –√(25 + 144)/25

= –√(169/25)

= -13/5

sin x = 1/cosec x

= 1/(-13/5)

= -5/13

cos x = – √(1 – sin² x)

= – √(1 – (-5/13)²)

= – √(169 - 25)/169

= – √(144/169)

= -12/13

sec x = 1/cos x

= 1/(-12/13)

= -13/12

∴ sin x = -5/13, cos x = -12/13, tan x = 5/12, cosec x = -13/5, sec x = -13/12

(ii) Since, in second quadrant, sin x and cosec x are positive. tan x, cot x, cos x, sec x are negative.

On using the formulas,

sin x = √(1 – cos² x)

= √(1 – (-1/2)²)

= √(4 - 1)/4

= √(3/4)

= √3/2

tan x = sin x/cos x

= (√3/2)/(-1/2)

= -√3

cot x = 1/tan x

= 1/-√3

= -1/√3

cosec x = 1/sin x

= 1/(√3/2)

= 2/√3

sec x = 1/cos x

= 1/(-1/2)

= -2

∴ sin x = √3/2, tan x = -√3, cosec x = 2/√3, cot x = -1/√3 sec x = -2

(iii) Since, in third quadrant, tan x and cot x are positive. sin x, cos x, sec x, cosec x are negative.

On using the formulas,

sin x = √(1 – cos² x)

= – √(1 - (-4/5)²)

= – √(25 - 16)/25

= – √(9/25)

= – 3/5

cos x = 1/sec x

= 1/(-5/4)

= -4/5

cot x = 1/tan x

= 1/(3/4)

= 4/3

cosec x = 1/sin x

= 1/(-3/5)

= -5/3

sec x = -√(1 + tan² x)

= – √(1 + (3/4)²)

= – √(16 + 9)/16

= – √ (25/16)

= -5/4

∴ sin x = -3/5, cos x = -4/5, cosec x = -5/3, sec x = -5/4, cot x = 4/3

(iv) Since, in first quadrant, all trigonometric ratios are positive.

Therefore, by using the formulas,

tan x = sin x/cos x

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

= 3/4

cosec x = 1/sin x

= 1/(3/5)

= 5/3

cos x = √(1 - sin² x)

= √(1 – (-3/5)²)

= √(25 - 9)/25

= √(16/25)

= 4/5

sec x = 1/cos x

= 1/(4/5)

= 5/4

cot x = 1/tan x

= 1/(3/4)

= 4/3

Thus, cos x = 4/5, tan x = 3/4, cosec x = 5/3, sec x = 5/4, cot x = 4/3