Question

Find the trigonometric functions of :

−45°

Answer 1

Angle of measure (– 45°):

Let m∠XOA = −45°

Its terminal arm (ray OA) intersects the standard unit circle at P(x, y).

Draw seg PM perpendicular to the X-axis.

∴ ΔOMP is a 45° – 45° – 90° triangle

 OP = 1,

OM = `1/sqrt(2)"OP"`

= `1/sqrt(2)(1)`

= `1/sqrt(2)`

PM = `1/sqrt(2)"OP"`

= `1/sqrt(2)(1)`

= `1/sqrt(2)`

Since point P lies in the 4^th quadrant,

x > 0, y < 0

∴ x = OM = `1/sqrt(2)` and y = - PM = `(-1)/sqrt(2)`

∴ P ≡ `(1/sqrt(2), (-1)/sqrt(2))`

sin (– 45°) = y = `- 1/sqrt(2)`

cos (– 45°) = x = `1/sqrt(2)`

tan (– 45°) = `y/x = ((-1/sqrt(2)))/((1/sqrt(2)))` = – 1

cosec (– 45°) = `1/y = 1/((-1/sqrt(2))) = -sqrt(2)`

sec (– 45°) = `1/x = 1/((1/sqrt(2))) = sqrt(2)`

cot (– 45°) = `x/y = ((1/sqrt(2)))/((-1/sqrt(2))` = – 1