Question

Find the trigonometric functions of :

−60°

Answer 1

Trigonometric Functions of (- 60°):

Let the measure of ∠XOA in a standard position be − 60°.

Its terminal arm (ray OA) intersects the standard unit circle in P (x, y), which lies in the fourth quadrant.

Draw segment PM perpendicular to the X-axis.

Then OM = |x| and MP = |y|.

In right-angled triangle OMP,

m∠MOP = 60° and OP= 1

m∠MPO = 30°

∴ OM = `1/2"OP" = 1/2 xx 1 = 1/2`.

∴ y² = MP² = OP² – OM²

= `1^2 - (1/2)^2`

`= 1 - 1/4` 

`= 3/4`

∴ y = `± sqrt(3)/2 and x = ±1/2`

But P lies in the fourth quadrant.

∴ x > 0 and y < 0

∴ x = `1/2 and y =  -sqrt(3)/(2)`

∴ P is `(1/2, - sqrt(3)/2)`

∴ sin (− 60°) = y = `-sqrt(3)/2`

cos (− 60°) = x = `1/2`

tan (− 60°) = `y/x = ((-(sqrt3)/2))/((1/2)) = -sqrt(3)`

cosec (− 60°) = `1/y = 1/((-(sqrt(3))/2)) = -2/sqrt(3)`

sec (− 60°) = `1/x = 1/((1/2))` = 2

cot (− 60°) = `x/y = ((1/2))/((-(sqrt(3))/2)) = -1/sqrt(3)`.