Question 1: If
and
, where
, find the values of the following: i)
ii)
iii)
iv)
Answer:
Given
and
, where
This means that A is in Quadrant I and B is in Quadrant I
Therefore,
Similarly,
i)
ii)
iii)
iv)
Question 2:
a) If
and
, where
, find the values of the following: i)
ii)
b) If
and
, where
and
both lie in Q II find the value of
Answer:
a) Given
and
, where
This means that A is in Quadrant II and B is in Quadrant I
Therefore,
Similarly,
i)
ii)
b) Given
and
, where
and
both lie in Q II
Therefore,
Similarly,
Question 3: If
and
, where
,
, find the values of the following: i)
ii)
Answer:
Given
and
, where
,
This means that A is in Quadrant III and B is in Quadrant IV
Therefore,
Similarly,
i)
ii)
Question 4: If
and
, where
and
, find
Answer:
Given
and
, where
and
This means that A is in Quadrant III and B is in Quadrant I
Therefore,
Question 5: If
and
, where
and
, find
Answer:
Given
and
, where
and
This means that A is in Quadrant II and B is in Quadrant IV
Therefore,
Similarly,
Similarly,
Question 6: If
and
, where
and
, find the following: i)
ii)
Answer:
Given
and
, where
and
This means that A is in Quadrant II and B is in Quadrant I
Therefore,
Similarly,
Similarly,
i)
ii)
Question 7: Evaluate the following:
i) ii)
iii) iv)
Answer:
i)
ii)
iii)
iv)
Question 8: If
and
, where A lies in the second quadrant and B in the third quadrant, find the values of the following:
i) ii)
iii)
Answer:
Given
and
, where A lies in the second quadrant and B in the third quadrant
Therefore,
i)
ii)
Question 9: Prove that
Answer:
LHS
RHS. Hence proved.
Question 10: Prove that
Answer:
LHS
RHS. Hence proved.
Question 11:
i)
ii)
iii)
Answer:
i)
ii)
iii)
Question 12:
i)
ii)
iii)
Answer:
i) LHS
Note:
RHS. Hence proved.
ii) LHS
Note:
RHS. Hence proved.
iii) LHS
Note:
RHS. Hence proved.
Question 13: Prove that:
Answer:
LHS
Note: Since
RHS. Hence proved.
Question 14: i) If
and
, prove that
ii) If
and
, prove that
Answer:
i)
ii)
Question 15: Prove that
i)
ii)
Answer:
i) LHS =
Lets calculate
Substituting it back
RHS. Hence proved.
ii) LHS
Since:
RHS. Hence Proved.
Question 16: Prove that:
i)
ii)
iii)
iv)
v)
vi)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
iv) RHS
LHS. Hence Proved
v) LHS
RHS. Hence Proved.
vi) LHS
RHS. Hence proved.
Question 17: Prove that:
i)
ii)
iii)
iv)
Answer:
i)
ii)
iii)
iv)
Question 18: Prove that
Answer:
RHS
LHS. Hence proved.
Question 19:
. show that
Answer:
Using Componendo and Dividendo
Question 20: If , prove that
Answer:
Now
Hence proved.
Question 21: If and
, find the value of
and
Answer:
Given and
Similarly,
Question 22: If and
, prove that
Answer:
Given and
Hence proved.
Question 23: If and
, prove that:
Answer:
Given and
Question 24: If lies in the first quadrant and
, then prove that
Answer:
Given
Question 25: If
, then prove that
Answer:
Given
. Hence proved.
Question 26: If and
, where
, then find the values of
and
Answer:
Given:
… … … i)
… … … ii)
Solving i) and ii) we get
Also
Question 27: If and
are two different values of
lying between
and
which satisfy the equation
, find the value of
Answer:
Given
Eliminating
Squaring both sides
If and
are roots of equation
Similarly
Eliminating
Squaring both sides
If and
are roots of equation
Now
Question 28: If and
, show that
i)
ii)
Answer:
Given, and
Similarly,
Question 29: Prove that:
i)
ii)
iii)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
Question 30: If , prove that
Answer:
Given
Hence Proved.
Question 31: If and
, show that
Answer:
Given and
. Hence proved.
Question 32: If angle is divided into two parts such that the tangents of one parts is
times the tangent of other, and
is their difference, then show that
Answer:
Let and
be the two parts
Applying componendo and dividendo
Hence proved.
Question 33: If
, then show that
Answer:
Given
Dividing both numerator and denominator by
Hence proved.
Question 34: If