**Evaluate the Trig Functions with Trig Ratios :**

Here we are going to see some example problems to evaluate the given trigonometric functions by trigonometric ratios.

**Question 1 :**

If cos θ : sin θ = 1 : 2, then find the value of (8 cos θ - 2 sin θ)/(4 cos θ + 2 sin θ)

**Solution :**

**Given that :**

**cos θ : sin θ = 1 : 2**

**cos θ/sin θ = 1/2**

**cot**** θ = 1/2 = Adjacent side / Opposite side**

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

= 1^{2} + 2^{2}

(Hypotenuse side)^{2} = 5

Hypotenuse side = √5

sin θ = 1/√5, cos θ = 2/√5

(8 cos θ - 2 sin θ)/(4 cos θ + 2 sin θ)

= (8/√5) - (4/√5)/(4/√5) + 2(2/√5)

= (4/√5)/(8/√5)

= 4/8

= 1/2

**Question 2 :**

From the given figure, prove that θ + Φ = 90°.Also prove that there are two other right angled triangles. Find sin a, cos β and tan Φ.

**Solution :**

In triangle ACD,

sin a = Opposite side / Hypotenuse side

sin a = DC/AC

sin a = 12/15 = 4/5

In triangle BDC,

cos β = Adjacent side / Hypotenuse side

= DB/BC

cos β = 16/20 = 4/5

tan Φ = Opposite side / Adjacent side

tan Φ = DB/DC = 16/12

= 4/3

**Question 3 :**

A boy standing at a point O finds his kite flying at a point P with distance OP = 25 m. It is at a height of 5 m from the ground. When the thread is extended by 10 m from P, it reaches a point Q. What will be the height QN of the kite from the ground? (use trigonometric ratios)

**Solution :**

In triangle OPM,

tan θ = PM/OP

= 5/25 ---(1)

tan θ = ON / (OP + PQ)

= h/(25 + 10)

= h/35 ---(2)

(1) = (2)

5/25 = h/35

(1/5)(35) = h

h = 7

After having gone through the stuff given above, we hope that the students would have understood, "Evaluate the Trig Functions with Trig Ratios"

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