Question

Find the trigonometric function of :

210°

Answer 1

Trigonometric Functions of 210° :

Let measure of ∠XOA in standard position be 210°.

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

Draw segment PM perpendicular to the X-axis.

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

Now, ΔOMP is a right-angled triangle in which m∠MOP = 30° and OP = 1.

∴ MP = `1/2"OP" = 1/2 xx 1 = 1/2`.

∴ |y| = `1/2`

By the distance formula,

x² + y² = 1

∴ `x^2 + (1/2)^2` = 1

∴ `x^2 + 1/4` = 1

∴ x² = `1 - 1/4 = 3/4`

∴ x = `± sqrt(3)/2  and y = ±1/2`

But P lies in the third quadrant.

∴ x < 0 and y < 0

∴ x = `-sqrt(3)/2, y = -1/2`

∴ P is `((-sqrt(3))/2, (-1)/2)`

∴ sin 210° = y = `(-1)/2`

cos 210° = x = `(-sqrt(3))/2`

tan 210° = `y/x = ((-1/2))/((-(sqrt(3))/2)) = 1/sqrt(3)`

cosec 210° = `1/y = (1)/((-1/2))` = – 2

sec 210° =  `1/x = 1/((-(sqrt(3))/2)) = -2/sqrt(3)`

cot 210° = `x/y = ((-(sqrt(3))/2))/((-1/2)) = sqrt(3)`.