Question

Answer the following:

Find the trigonometric functions of :

315°

Answer 1

Trigonometric Functions of 315° :

Let measure of ∠XOA in standard position be 315°

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 = l x l and MP = l y l.

In right-angled triangle OMP,

m∠MOP = 45° and OP = 1

∴ OM = MP

∴ | x | = | y |

By the distance formula,

x² + y² = 1

∴ x² + y² = 1

∴ 2x² = 1

∴ x² = `1/2`

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

But P lies in the fourth quadrant,

∴ x > 0 and y < 0

∴ x = ` 1/sqrt(2) and y = -1/sqrt(2)`

∴ P is `(1/sqrt(2), -1/sqrt(2))`

∴ sin 315° = y = `-1/sqrt(2)`

cos 315° = x = `1/sqrt(2)`

tan 315° = `y/x = ((-(1)/2))/((1/sqrt(2))` = – 1

cosec 315° = `1/y = 1/((-(1)/sqrt(2))) = -sqrt(2)`

sec 315° = `1/x = 1/((1/sqrt(2))) = sqrt(2)`

cot 315° = ` x/y = ((1/sqrt(2)))/((-(1)/sqrt(2)))` = – 1

[Note: Answer given in the textbook of cot 315° is 1. However, as per our calculation it is –1.]