Question 1: Find the maximum and minimum values of each of the following trigonometrical expressions:

i) ii)

iii) iv)

Answer:

i) Let

We know that

Hence minimum and maximum value of are and respectively.

ii) Given

Let

We know that

Hence minimum and maximum value of are and respectively.

iii) Let

We know

Hence minimum and maximum value of are and respectively.

iv) Given

Let

We know that

Hence minimum and maximum value of are and respectively.

Question 2: Reduce each of the following expressions to the sine and cosine of a single expression: i) ii) iii)

Answer:

i)

Multiplying and dividing by we get

Also

ii)

Multiplying and dividing by we get

Similarly,

iii)

Multiplying and dividing by we get

Let and

where

Similarly,

Let and

where

Question 3: Show that is positive

Answer:

Given

which is a positive number as is in first quadrant.

Question 4: Prove that lies between and

Answer:

Given

Assume and

Dividing and multiplying by we get

Assume we get ,

We know that the max. and minimum value of any cosine is and

We know,

We know because value of is more that

So we replace with . The above inequality still holds true.

So range of above expression can be