**Quotient relation of trigonometric ratios : **

When we divide the trigonometric ratio sinθ by cosθ, the quotient is tanθ.

When we divide the trigonometric ratio cosθ by sinθ, the quotient is cotθ.

When we divide the trigonometric ratio cscθ by secθ, the quotient is cotθ.

When we divide the trigonometric ratio secθ by cscθ, the quotient is tanθ.

The above said divisions are called as quotient relation of trigonometric ratios.

In the triangle above, according SOHCAHTOA, we have

sinθ = opposite side / hypotenuse = BC / AC

cosθ = adjacent side / hypotenuse = AB / AC

Now, let us divide sinθ by cosθ

sinθ / cosθ = (BC/AC) ÷ (AB/AC)

sinθ / cosθ = (BC/AC) x (AC/AB)

sinθ / cosθ = BC /AB = tanθ

(Because, tanθ = opposite side / adjacent side = BC / AB)

Therefore,

sinθ / cosθ = tanθ

Now, let us divide cosθ by sinθ

cosθ / sinθ = (AB/AC) ÷ (BC/AC)

cosθ / sinθ = (AB/AC) x (AC/BC)

cosθ / sinθ = AB / BC = cotθ

(Because, cotθ = adjacent side / opposite side = AB / BC)

Therefore,

cosθ / sinθ = cotθ

cscθ = 1 / sinθ = AC / BC

secθ = 1 / cosθ = AC / AB

Now, let us divide cscθ by secθ

cscθ / secθ = (AC/BC) ÷ (AC/AB)

cscθ / secθ = (AC/BC) x (AB/AC)

cscθ / secθ = AB / BC = cotθ

(Because, cotθ = adjacent side / opposite side = AB / BC)

Therefore,

cscθ / secθ = cotθ

Now, let us divide secθ by cscθ

secθ / cscθ = (AC/AB) ÷ (AC/BC)

secθ / cscθ = (AC/AB) x (BC/AC)

secθ / cscθ = BC / AB = tanθ

(Because, tanθ = opposite side/adjacent side = BC / AB)

Therefore,

secθ / cscθ = tanθ

**Problem 1 :**

In the right triangle PQR given below, find the value of sinθ and cosθ. Using them, find the value of tanθ and cotθ

**Solution :**

From the figure given above,

opposite side = 5

adjacent side = 12

hypotenuse = 13

Therefore,

sinθ = PQ/RQ = 5/13

cosθ = PR/RQ = 12/13

tanθ = sinθ / cosθ = (5/13) ÷ (12/13)

tanθ = (5/13) x (13/12)

**tanθ = 5/12**

cotθ = cosθ / sinθ = (12/13) ÷ (5/13)

cotθ = (12/13) x (13/5)

**cotθ = 12/5**

**Problem 2 :**

From the figure given below, find the value of sinθ and cosθ. Using them, find the value of tanθ and cotθ

**Solution : **

From the figure given above, AC = 24 and BC = 7.

By Pythagorean theorem,

AB² = BC² + CA²

AB² = 7² + 24²

AB² = 49 + 576

AB² = 625

AB² = 25²

AB = 25

Now, we can use the three sides to find the six trigonometric ratios of angle θ.

Therefore,

opposite side = 7

adjacent side = 24

hypotenuse = 25

Therefore,

sinθ = BC/AB = 7/25

cosθ = AC/AB = 24/25

tanθ = sinθ / cosθ = (7/25) ÷ (24/25)

tanθ = (7/25) x (25/24)

**tanθ = 7/24**

sinθ = BC/AB = 7/25

cosθ = AC/AB = 24/25

cotθ = cosθ / sinθ = (24/25) ÷ (7/25)

cotθ = (24/25) x (25/7)

**cotθ = 24/7**

After having gone through the stuff given above, we hope that the students would have understood "Quotient relation of trigonometric ratios"

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