Question 1: If in a and ; find the ratios of their sides.

Answer:

Given in a and ;

Using Sine Rule

Question 2: If in a , then find .

Answer:

Given in a

Using sine rule,

Question 3: If in , if and and , then find and

Answer:

Given in , if and and

Using sine rule,

In any triangle ABC, prove the following:

Question 4:

Answer:

LHS

RHS. Hence proved.

Question 5:

Answer:

LHS

RHS. Hence proved.

Question 6:

Answer:

LHS

Dividing Numerator and Denominator by we get

RHS. Hence proved.

Question 7:

Answer:

LHS

Dividing Numerator and Denominator by we get

RHS. Hence proved.

Question 8:

Answer:

LHS

RHS. Hence proved.

Question 9:

Answer:

LHS

RHS. Hence proved.

Question 10:

Answer:

LHS

RHS. Hence proved.

Question 11:

Answer:

Using cosine rule

RHS

LHS. Hence proved.

Question 12:

Answer:

LHS

RHS. Hence proved.

Question 13:

Answer:

LHS

LHS. Hence proved.

Question 14:

Answer:

LHS

RHS. Hence proved.

Question 15:

Answer:

LHS

RHS. Hence proved.

Question 16:

Answer:

LHS

RHS. Hence proved.

Question 17:

Answer:

LHS

RHS. Hence proved.

Question 18:

Answer:

LHS

RHS. Hence proved.

Question 19:

Answer:

LHS

RHS. Hence proved.

Question 20:

Answer:

LHS

RHS. Hence proved.

Question 21:

Answer:

LHS

Similarly.

and

Hence proved.

Question 22:

Answer:

LHS

RHS. Hence proved.

Similarly,

Question 23:

Answer:

Similarly,

Question 24:

Answer:

LHS

RHS. Hence proved.

Question 25: In prove that, if be any angle, then

Answer:

RHS

LHS. Hence proved.

Question 26: In , if , show that the triangle is a right angled.

Answer:

Let , and

Given

Since the triangle satisfies Pythagoras Theorem, the triangle is a right angles triangle.

Question 27: In any , if are in AP, prove that and are also in AP.

Answer:

Given are in AP

are in AP

are in AP

are in AP

are in AP

are in AP

are in AP

are in AP

are in AP

are in AP

Question 28: The upper part of a tree broken over by the wind makes an angle of with the ground and the distance from the root to the point where the top of the tree touches the ground is m. Using sine rule, find the height of the tree.

Answer:

Using sine rule,

m

Question 29: At the foot of the mountain the elevation of the peak is , after ascending m towards the mountain up the slope of 30^o inclination, the elevation is found to be . Find the height of the mountain.

Answer:

m

Let m

In

m

Therefore the height of triangle is m

Question 30: A person observes the angle of elevation of the peak of a hill from a station to be . He walks c meters along a slope inclined at an angle and finds the angle of elevation of the peak of the hill as . Show that the height of the peak above the ground is

Answer:

The person is observing the peak from point

Distance traveled is and the angle of inclination is

Observing the peak from point , angle of inclination is

Now consider

… … … … … i)

Now consider

We know,

In

Now consider ,

Substituting the value of from i)

… … … … … ii)

We need to find the total height of the peak

… … … … … iii)

From ,

… … … … … iv)

From

… … … … … v)

Hence proved.

Question 31: If the sides of are in HP, prove that are in HP too.

Answer:

If sides are in HP

are in AP

Using sine rule,

are in AP

are in HP