**Q.1 Â Â In , right angled at B, AB = 24 cm, BC = 7 cm. Determine :**

**Â Â Â Â Â Â (i) sin A, cos A Â Â Â Â Â Â Â (ii) sin C, cos C**

* Sol. Â Â Â Â *Let us draw a right .

Â Â Â Â Â Â Â By using the Pythagoras theorem, we have

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 576 + 49 = 625

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

**Q.2 Â Â In adjoining figure, find tan P â€“ cot R.
**

Â Â Â Â Â Â

Â Â Â Â Â Â

Â Â Â Â Â Â

Â Â Â Â Â Â

Â Â Â Â Â Â Therefore, and

Â Â Â Â Â Â Hence, tan P â€“ cot R =

**Q.3 Â Â If calculate cos A and tan A.**

* Sol.*Â Â Â Â Consider a in which

Â Â Â Â Â Â Â For we have

Â Â Â Â Â Â Â Base = AB, Perp = BC and Hyp = AC

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â Let BC = 3k and AC = 4k.

Â Â Â Â Â Â Â Then,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â

**Q.4 Â Â Given 15 cot A = 8, find sin A and sec A.**

* Sol.*Â Â Â Â Consider a in which

Â Â Â Â Â Â Â For , we have

Â Â Â Â Â Â Â Base = AB, Perp = BC

Â Â Â Â Â Â Â and Hyp = AC

Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â [Since, 15 cot A = 8 ]

Â Â Â Â Â Â Â Let AB = 8k and BC = 15k.

Â Â Â Â Â Â Â Then,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â and,

**Q.5 Â Â GivenÂ , Â calculate all other trigonometric ratios.Â **

Â Â Â Â Â Â Â Then, Base = AB, Perp = BC and Hyp = AC

Â Â Â Â Â Â Â Therefore, Â

Â Â Â Â Â Â Â Let AC = 13k and AB = 12k. Then,

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â

**Q.6 Â Â Â If and are acute angles such that cos A = cos B, then show that
**

Â Â Â Â Â Â and,

Â Â Â Â Â Â But, cos A = cos B [Given]

Â Â Â Â Â Â

Â Â Â Â Â Â

Â Â Â Â Â Â Since, in , AC = BC

Â Â Â Â Â Â [Angles opposite to equal sides are equal]

**Q.7 Â Â If , evaluate :**

**Â Â Â Â Â Â (i) **

**Â Â Â Â Â Â (ii) **

* Sol.*Â Â Â Â Consider a in which Â and Then, Base = AB, Perp = BC and Hyp = AC

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â Let AB = 7k and BC = 8k.

Â Â Â Â Â Â Â Then,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â and

Â Â Â Â Â Â Â **(i)** Therefore,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â **(ii)**

**Q.8 Â Â If 3 cot A = 4, check whether or not.
**

Â Â Â Â Â Â Â For , we have

Â Â Â Â Â Â Â Base = AB, Perp = BC and Hyp = AC

Â Â Â Â Â Â Â Therefore, Â Â Â Â

Â Â Â Â Â Â Â Let AB = 4k and BC = 3k [3cot A = 4 ]

Â Â Â Â Â Â Â Then,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â and,

Â Â Â Â Â Â Â L.H.S.

Â Â Â Â Â Â Â R.H.S. =

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â L.H.S = R.H.S.

Â Â Â Â Â Â Â Therefore,

**Q.9 Â Â In right angled at B, if tan A = , find the value of**

**Â Â Â Â Â Â (i) sin A cos C + cos A sin C**

**Â Â Â Â Â Â (ii) cos A cos C â€“ sin A sin C**

* Sol.*Â Â Â Â Consider a , in which .

Â Â Â Â Â Â Â For , we have

Â Â Â Â Â Â Â Base = AB, Perp = BC

Â Â Â Â Â Â Â and Hyp = AC

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â =

Â Â Â Â Â Â Â Let BC = k and AB = k .

Â Â Â Â Â Â Â Then,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â =

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â For , we have

Â Â Â Â Â Â Â Base = BC, Perp = AB and Hyp = AC

Â Â Â Â Â Â Â Therefore,

Â Â Â Â Â Â Â and,

Â Â Â Â Â Â Â **(i)** sinA cosC + cosA sinC =

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â **(ii)** cos A cosC â€“ sin A sin C =

**Q.10 Â Â In , right angled at Q, PR + QR 25 cm and PQ = 5 cm. Determine the values of sin P cos P and tan P.
**

Â Â Â Â Â Â Â Â PR + QR = 25 cm and PQ = 5cm

Â Â Â Â Â Â Â Â Let QR = x cm

Â Â Â Â Â Â Â Â Therefore, PR = (25 â€“ x) cm

Â Â Â Â Â Â Â Â By Pythagoras theorem, we haveÂ

Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Therefore, RQ = 12cm

Â Â Â Â Â Â Â Â RP = (25 â€“ 12)cm = 13cm

Â Â Â Â Â Â Â Â Now,

Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â and

**Q.11 Â Â State whether the following are true or false. Justify your answer.**

**Â Â Â Â Â Â Â (i) The value of tan A is always less than 1.**

**Â Â Â Â Â Â Â (ii) for some value of angle A .**

**Â Â Â Â Â Â Â (iii) cos A is the abbreviation used for the cosecant of angle A.**

**Â Â Â Â Â Â Â (iv) cot A is the product of cot and A.**

**Â Â Â Â Â Â Â (v) for some angle .**

* Sol.*Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â

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