1. In ABC, right angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i)
(ii)
1. In ABC, right angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i)
(ii)
Explanation:
Let's create a triangle with a right angle at B, ABC.
Making use of Pythagoras' Theorem
= = 576 + 49 = 625
AC = 25 cm
(i) ,
(ii) ,
2. In the given figure, find tan P – cot R.
2. In the given figure, find tan P – cot R.
Explanation :
Making use of Pythagoras' Theorem
=169 – 144 = 25
QR = 5 cm
= = = 0.
3. If calculate and
3. If calculate and
Explanation:
Presented: An ABC triangle, where B =
Let BC = and AC =
Then, Making use of Pythagoras' Theorem
AB = =
= =
4. Given find and
4. Given find and
Explanation:
Presented: An ABC triangle, where, B =
Let AB = and BC =
Making use of Pythagoras' Theorem
AC =
=
=
= =
5. Given calculate all other trigonometric ratios.
5. Given calculate all other trigonometric ratios.
Explanation:
Think of a triangle ABC where, A = and B =
Let AB = and BC =
Making use of Pythagoras' Theorem
BC =
=
=
= =
6. If And B is acute angles such that then show that A = B.
6. If And B is acute angles such that then show that A = B.
Explanation:
In the ABC right triangle,
and
But [Given]
AC = BC
A = B
[Angles opposing equal sides have the same value.]
7. If evaluate:
(i)
(ii)
7. If evaluate:
(i)
(ii)
Explanation:
Think of a triangle ABC where, A = and B =
Let AB = and BC =
Making use of Pythagoras' Theorem
AC =
=
=
= =
(i) =
= = =
(ii) = =
8. If check whether or not.
8. If check whether or not.
Explanation:
Think of a triangle ABC where B =
And
Let us take AB = and BC =
Making use of Pythagoras' Theorem
AC =
=
=
= =
And
Here, L.H.S. =
= =
R.H.S. =
= =
L.H.S. = R.H.S.
=
9. In ABC right angles at B, if find a value of:
(i)
(ii)
9. In ABC right angles at B, if find a value of:
(i)
(ii)
Explanation:
Think of a triangle ABC where B =
Let BC = and AB =
Making use of Pythagoras' Theorem
AC =
=
= = =
For the C, Base = BC, Perpendicular = AB and Hypotenuse = AC
(i) =
= = 1
(ii) =
=
10. In PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P, and tan P.
10. In PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P, and tan P.
Explanation:
In PQR, right-angled at Q.
PR + QR = 25 cm and PQ = 5 cm
Let QR = cm and PR = cm
Making use of Pythagoras' Theorem
RQ = 12 cm and RP = 25 – 12 = 13 cm
And
11. State whether the following are true or false. Justify your answer.
(i) The value of is always less than 1.
(ii) for some value of angle A.
(iii) is the abbreviation used for the cosecant of angle A.
(iv) is the product of and A.
(v) for some angle
11. State whether the following are true or false. Justify your answer.
(i) The value of is always less than 1.
(ii) for some value of angle A.
(iii) is the abbreviation used for the cosecant of angle A.
(iv) is the product of and A.
(v) for some angle
Explanation:
(i) False A right triangle's sides can have any length, hence their values can be anything.
(ii) True as has to be bigger than 1.
(iii) False as is a shorthand for cosine A.
(iv) False as is not the result of A and "cot." When A is separated, the word "cot" has no meaning.
(v) False as not exceeding 1.
12. Evaluate the following:
(i)
(ii)
(iii)
(iv)
(v)
12. Evaluate the following:
(i)
(ii)
(iii)
(iv)
(v)
Explanation:
(i)
=
= =1
(ii)
=
= =2
(iii)
= =
= =
=
= =
=
(iv)
= = =
=
= =
(v)
=
=
= =
13. Choose the correct option and justify your choice:
(i) =
(A)
(B)
(C)
(D)
(ii) =
(A)
(B) 1
(C)
(D) 0
(iii) is true when A =
(A)
(B)
(C)
(D)
(iv) =
(A)
(B)
(C)
(D) None of these
13. Choose the correct option and justify your choice:
(i) =
(A)
(B)
(C)
(D)
(ii) =
(A)
(B) 1
(C)
(D) 0
(iii) is true when A =
(A)
(B)
(C)
(D)
(iv) =
(A)
(B)
(C)
(D) None of these
Explanation:
(i) (A)
=
= = =
(ii) (D) = = = 0
(iii) (A) As A = 0, then
and
=
when A = 0
(iv) (C)
= =
= =
14. If and find A and B.
14. If and find A and B.
Explanation:
A + B = ……….(i)
A – B = ……….(ii)
Equations (i) and (ii) together give us,
2A = A =
When we take away equations (i) and (ii), give us
2B = B =
15. State whether the following are true or false. Justify your answer.
(i)
(ii) The value of increases as increases.
(iii) The value of increases as increases.
(iv) for all values of
(v) is not defined for
15. State whether the following are true or false. Justify your answer.
(i)
(ii) The value of increases as increases.
(iii) The value of increases as increases.
(iv) for all values of
(v) is not defined for
Explanation:
(i) False, because
= 1
And =
(ii) True, because
It is obvious that as increases, the value of also rises.
(iii) False, because
It is evident that as increases, the value of diminishes.
(iv) False as it is only true for
(v) True, because and
= i.e. undefined.
16. Evaluate:
(i)
(ii)
(iii)
(iv)
16. Evaluate:
(i)
(ii)
(iii)
(iv)
Explanation:
(i) =
= = 1
(ii) =
= = 1
(iii)
=
= = 0
(iv)
=
= =0
17. Show that:
(i)
(ii)
17. Show that:
(i)
(ii)
Explanation:
(i)L.H.S.
=
=
= = 1 = R.H.S.
(ii) R.H.S.
=
= = 0 = R.H.S.
18. If where 2A is an acute angle, find the value of A.
18. If where 2A is an acute angle, find the value of A.
Explanation:
Presented:
A =
19. If proven that
19. If proven that
Explanation:
Presented:
A + B =
20. If where 4A is an acute angle, find the value of A.
20. If where 4A is an acute angle, find the value of A.
Explanation:
Presented:
A =
21. If A, B and C are interior angles of an ABC, then show that
21. If A, B and C are interior angles of an ABC, then show that
Explanation:
Presented: A, B and C are interior angles of an ABC.
A + B + C =
22. Express in terms of trigonometric ratios of angles between and
22. Express in terms of trigonometric ratios of angles between and
Explanation:
=
=
23. Express the trigonometric ratios and in terms of
23. Express the trigonometric ratios and in terms of
Explanation:
For
utilizing identity
For
utilizing identity
=
For
24. Write the other trigonometric ratios of ∠A in terms of
24. Write the other trigonometric ratios of ∠A in terms of
Explanation:
For
utilizing identity,
=
For
For
utilizing identity
For
=
For
25. Evaluate:
(i)
(ii)
25. Evaluate:
(i)
a(ii)
Explanation:
(i)
=
=
=
(ii)
=
=
= = 1
26. Choose the correct option. Justify your choice:
(i) =
(A) 1
(B) 9
(C) 8
(D) 0
(ii) =
(A) 0
(B) 1
(C) 2
(D) none of these
(iii) =
(A)
(B)
(C)
(D)
(iv) =
(A)
(B)
(C)
(D) none of these
26. Choose the correct option. Justify your choice:
(i) =
(A) 1
(B) 9
(C) 8
(D) 0
(ii) =
(A) 0
(B) 1
(C) 2
(D) none of these
(iii) =
(A)
(B)
(C)
(D)
(iv) =
(A)
(B)
(C)
(D) none of these
Explanation:
(i) (B)
=
=
(ii) (C)
=
=
=
=
=
= = 2
(iii)(D)
=
=
= =
=
(iv)(D) =
= =
= =
27. Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
(i)
(ii)
(iii)
(iv)
(v) using the identity
(vi)
(vii)
(viii)
(ix)
(x)
27. Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
(i)
(ii)
(iii)
(iv)
(v) using the identity
(vi)
(vii)
(viii)
(ix)
(x)
Explanation:
(i) L.H.S.
=
=
=
=
=
=
=
= = R.H.S.
(ii) L.H.S.
=
=
=
= =
= = = R.H.S
(iii) L.H.S.
=
=
=
=
=
=
=
= =
=
(iv) L.H.S.
= =
=
=
= = R.H.S.
(v) L.H.S.
dividing up every term by
= =
=
=
=
= = R.H.S.
(vi) L.H.S.
=
=
=
= =
= = R.H.S.
(vii) L.H.S.
=
=
=
= =
= = R.H.S
(viii) L.H.S.
=
=
=
=
=
=
=
= R.H.S.
(ix) L.H.S.
=
=
= =
=
dividing up every term by, ,
=
=
= = R.H.S.
(x) L.H.S. =
= = = R.H.S.
We consider, Middle side = =
=
= =
= = R.H.S.
CHAPTER 8 INTRODUCTION TO TRIGONOMETRY