**Finding Trigonometric Ratios from the Given Triangle :**

In this section, you will learn how to find the values of trigonometric ratios from the given triangle.

**Question 1 :**

From the given figure, find all the trigonometric ratios of angle B.

**Solution :**

Here we have to mark the angle at B.

From the given triangle,

Hypotenuse side (BC) = 41

Opposite side (AC) = 9

Adjacent side (AB) = 40

sin B = Opposite side / Hypotenuse side

sin B = AC/BC = 6/41

cos B = Adjacent side / Hypotenuse side

cos B = AB/BC = 40/41

tan B = Opposite side / Adjacent side

tan B = AC/AB = 6/40

cosec B = Hypotenuse side/Opposite side

cosec B = BC/AC = 41/6

sec B = Hypotenuse side/Adjacent side

sec B = BC/AB = 41/40

cot B = Adjacent side / Opposite side

cot B = AB/AC = 40/6

**Question 2 :**

From the given figure, find the values of

(i) sin B (ii) sec B (iii) cot B

(iv) cos C (v) tan C (vi) cosec C

**Solution :**

In triangle ABD,

AB^{2} = AD^{2} + BD^{2}

13^{2} = AD^{2} + 5^{2}

169 - 25 = AD^{2}

AD^{2} = 144

AD = 12

In triangle ADC,

AC^{2} = AD^{2} + DC^{2}

AC^{2} = 12^{2} + 16^{2}

AC^{2} = 144 + 256

AC = 20

(i) sin B = AD/AB sin B = 12/13 (ii) sec B = AB/BD sec B = 13/5 (iii) cot B = BD/AD cot B = 5/12 |
(iv) cos C = DC/AC cos C = 16/20 = 4/5 (v) tan C = AD/DC tan C = 12/16 = 3/4 (vi) cosec C = AC/AD cosec C = 20/12 = 5/3 |

**Question 3 :**

If 2 cos θ = √3, then find all the trigonometric ratios of angle θ.

**Solution :**

cos θ = √3/2 = Adjacent side / Hypotenuse side

(Opposite side)^{2} = (Hypotenuse side)^{2} - (Adjacent side)^{2}

(Opposite side)^{2} = 2^{2} - (√3)^{2 } = 4 - 3

Opposite side = 1

sin A = Opposite side / Hypotenuse = 1/2

cos A = Adjacent side / Hypotenuse = √3/2

tan A = Opposite side / Adjacent side = 1/√3

cosec A = Hypotenuse / Opposite side = 2/1

sec A = Hypotenuse / Adjacent side = 2/√3

cot A = Adjacent side / Opposite side = √3/1

After having gone through the stuff given above, we hope that the students would have understood how to find trigonometric ratios from the given triangle.

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