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