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Mathematics · Textbook solutions

Introduction to Trigonometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 59 questions

8.2 Trigonometric Ratios

23 q

Solved Examples

Worked · 5
  1. 8.1 Eg.1
    Given tan⁡A=43\tan \text{A} = \dfrac{4}{3}, find the other trigonometric ratios of the angle A.
  2. 8.1 Eg.2
    If ∠\angleB and ∠\angleQ are acute angles such that sin⁡B=sin⁡Q\sin \text{B} = \sin \text{Q}, then prove that ∠\angleB == ∠\angleQ.
  3. 8.1 Eg.3
    Consider △\triangleACB, right-angled at C, in which AB =29= 29 units, BC =21= 21 units and ∠\angleABC =θ= \theta (see Fig. 8.10). Determine the values of (i) cos⁡2θ+sin⁡2θ\cos^2\theta + \sin^2\theta, (ii) cos⁡2θ−sin⁡2θ\cos^2\theta - \sin^2\theta
  4. 8.1 Eg.4
    In a right triangle ABC, right-angled at B, if tan⁡A=1\tan \text{A} = 1, then verify that 2sin⁡Acos⁡A=12 \sin \text{A} \cos \text{A} = 1.
  5. 8.1 Eg.5
    In △\triangleOPQ, right-angled at P, OP =7= 7 cm and OQ −- PQ =1= 1 cm (see Fig. 8.12). Determine the values of sin⁡Q\sin \text{Q} and cos⁡Q\cos \text{Q}.

Exercise 8.1

Practice · 18
  1. In △\triangleABC, right-angled at B, AB =24= 24 cm, BC =7= 7 cm. Determine :
    Ex 8.1 Q1 (i)
    sin⁡A\sin \text{A}, cos⁡A\cos \text{A}
  2. Ex 8.1 Q1 (ii)
    sin⁡C\sin \text{C}, cos⁡C\cos \text{C}
  3. Ex 8.1 Q2
    In Fig. 8.13, find tan⁡P−cot⁡R\tan \text{P} - \cot \text{R}.
  4. Ex 8.1 Q3
    If sin⁡A=34\sin \text{A} = \dfrac{3}{4}, calculate cos⁡A\cos \text{A} and tan⁡A\tan \text{A}.
  5. Ex 8.1 Q4
    Given 15cot⁡A=815 \cot \text{A} = 8, find sin⁡A\sin \text{A} and sec⁡A\sec \text{A}.
  6. Ex 8.1 Q5
    Given sec⁡θ=1312\sec\theta = \dfrac{13}{12}, calculate all other trigonometric ratios.
  7. Ex 8.1 Q6
    If ∠\angleA and ∠\angleB are acute angles such that cos⁡A=cos⁡B\cos \text{A} = \cos \text{B}, then show that ∠\angleA == ∠\angleB.
  8. Ex 8.1 Q8
    If 3cot⁡A=43 \cot \text{A} = 4, check whether 1−tan⁡2A1+tan⁡2A=cos⁡2A−sin⁡2A\dfrac{1 - \tan^2 \text{A}}{1 + \tan^2 \text{A}} = \cos^2 \text{A} - \sin^2 \text{A} or not.
  9. Ex 8.1 Q10
    In △\trianglePQR, right-angled at Q, PR ++ QR =25= 25 cm and PQ =5= 5 cm. Determine the values of sin⁡P\sin \text{P}, cos⁡P\cos \text{P} and tan⁡P\tan \text{P}.
  10. If cot⁡θ=78\cot\theta = \dfrac{7}{8}, evaluate :
    Ex 8.1 Q7 (i)
    (1+sin⁡θ)(1−sin⁡θ)(1+cos⁡θ)(1−cos⁡θ)\dfrac{(1 + \sin\theta)(1 - \sin\theta)}{(1 + \cos\theta)(1 - \cos\theta)}
  11. Ex 8.1 Q7 (ii)
    cot⁡2θ\cot^2\theta
  12. In triangle ABC, right-angled at B, if tan⁡A=13\tan \text{A} = \dfrac{1}{\sqrt{3}}, find the value of:
    Ex 8.1 Q9 (i)
    sin⁡Acos⁡C+cos⁡Asin⁡C\sin \text{A} \cos \text{C} + \cos \text{A} \sin \text{C}
  13. Ex 8.1 Q9 (ii)
    cos⁡Acos⁡C−sin⁡Asin⁡C\cos \text{A} \cos \text{C} - \sin \text{A} \sin \text{C}
  14. State whether the following are true or false. Justify your answer.
    Ex 8.1 Q11 (i)
    The value of tan⁡A\tan \text{A} is always less than 1.
  15. Ex 8.1 Q11 (ii)
    sec⁡A=125\sec \text{A} = \dfrac{12}{5} for some value of angle A.
  16. Ex 8.1 Q11 (iii)
    cos⁡A\cos \text{A} is the abbreviation used for the cosecant of angle A.
  17. Ex 8.1 Q11 (iv)
    cot⁡A\cot \text{A} is the product of cot⁡\cot and A.
  18. Ex 8.1 Q11 (v)
    sin⁡θ=43\sin\theta = \dfrac{4}{3} for some angle θ\theta.

8.3 Trigonometric Ratios of Some Specific Angles

18 q

Solved Examples

Worked · 3
  1. 8.2 Eg.6
    In △\triangleABC, right-angled at B, AB =5= 5 cm and ∠\angleACB =30∘= 30^\circ (see Fig. 8.19). Determine the lengths of the sides BC and AC.
  2. 8.2 Eg.7
    In △\trianglePQR, right-angled at Q (see Fig. 8.20), PQ =3= 3 cm and PR =6= 6 cm. Determine ∠\angleQPR and ∠\anglePRQ.
  3. 8.2 Eg.8
    If sin⁡(A−B)=12\sin(\text{A} - \text{B}) = \dfrac{1}{2}, cos⁡(A+B)=12\cos(\text{A} + \text{B}) = \dfrac{1}{2}, 0∘<A+B≤90∘0^\circ < \text{A} + \text{B} \le 90^\circ, A>B\text{A} > \text{B}, find A and B.

Exercise 8.2

Practice · 15
  1. Evaluate the following :
    Ex 8.2 Q1 (i)
    sin⁡60∘cos⁡30∘+sin⁡30∘cos⁡60∘\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ
  2. Ex 8.2 Q1 (ii)
    2tan⁡245∘+cos⁡230∘−sin⁡260∘2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ
  3. Ex 8.2 Q1 (iii)
    cos⁡45∘sec⁡30∘+cosec⁡30∘\dfrac{\cos 45^\circ}{\sec 30^\circ + \operatorname{cosec} 30^\circ}
  4. Ex 8.2 Q1 (iv)
    sin⁡30∘+tan⁡45∘−cosec⁡60∘sec⁡30∘+cos⁡60∘+cot⁡45∘\dfrac{\sin 30^\circ + \tan 45^\circ - \operatorname{cosec} 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}
  5. Ex 8.2 Q1 (v)
    5cos⁡260∘+4sec⁡230∘−tan⁡245∘sin⁡230∘+cos⁡230∘\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}
  6. Choose the correct option and justify your choice :
    Ex 8.2 Q2 (i)
    2tan⁡30∘1+tan⁡230∘=\dfrac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =
    1. A.
      sin⁡60∘\sin 60^\circ
    2. B.
      cos⁡60∘\cos 60^\circ
    3. C.
      tan⁡60∘\tan 60^\circ
    4. D.
      sin⁡30∘\sin 30^\circ
  7. Ex 8.2 Q2 (ii)
    1−tan⁡245∘1+tan⁡245∘=\dfrac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =
    1. A.
      tan⁡90∘\tan 90^\circ
    2. B.
      11
    3. C.
      sin⁡45∘\sin 45^\circ
    4. D.
      00
  8. Ex 8.2 Q2 (iii)
    sin⁡2A=2sin⁡A\sin 2\text{A} = 2 \sin \text{A} is true when A ==
    1. A.
      0∘0^\circ
    2. B.
      30∘30^\circ
    3. C.
      45∘45^\circ
    4. D.
      60∘60^\circ
  9. Ex 8.2 Q2 (iv)
    2tan⁡30∘1−tan⁡230∘=\dfrac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =
    1. A.
      cos⁡60∘\cos 60^\circ
    2. B.
      sin⁡60∘\sin 60^\circ
    3. C.
      tan⁡60∘\tan 60^\circ
    4. D.
      sin⁡30∘\sin 30^\circ
  10. Ex 8.2 Q3
    If tan⁡(A+B)=3\tan(\text{A} + \text{B}) = \sqrt{3} and tan⁡(A−B)=13\tan(\text{A} - \text{B}) = \dfrac{1}{\sqrt{3}}; 0∘<A+B≤90∘0^\circ < \text{A} + \text{B} \le 90^\circ; A>B\text{A} > \text{B}, find A and B.
  11. State whether the following are true or false. Justify your answer.
    Ex 8.2 Q4 (i)
    sin⁡(A+B)=sin⁡A+sin⁡B\sin(\text{A} + \text{B}) = \sin \text{A} + \sin \text{B}.
  12. Ex 8.2 Q4 (ii)
    The value of sin⁡θ\sin\theta increases as θ\theta increases.
  13. Ex 8.2 Q4 (iii)
    The value of cos⁡θ\cos\theta increases as θ\theta increases.
  14. Ex 8.2 Q4 (iv)
    sin⁡θ=cos⁡θ\sin\theta = \cos\theta for all values of θ\theta.
  15. Ex 8.2 Q4 (v)
    cot⁡A\cot \text{A} is not defined for A=0∘\text{A} = 0^\circ.

8.4 Trigonometric Identities

18 q

Solved Examples

Worked · 4
  1. 8.3 Eg.9
    Express the ratios cos⁡A\cos \text{A}, tan⁡A\tan \text{A} and sec⁡A\sec \text{A} in terms of sin⁡A\sin \text{A}.
  2. 8.3 Eg.10
    Prove that sec⁡A (1−sin⁡A)(sec⁡A+tan⁡A)=1\sec \text{A}\,(1 - \sin \text{A})(\sec \text{A} + \tan \text{A}) = 1.
  3. 8.3 Eg.11
    Prove that cot⁡A−cos⁡Acot⁡A+cos⁡A=cosec⁡A−1cosec⁡A+1\dfrac{\cot \text{A} - \cos \text{A}}{\cot \text{A} + \cos \text{A}} = \dfrac{\operatorname{cosec} \text{A} - 1}{\operatorname{cosec} \text{A} + 1}.
  4. 8.3 Eg.12
    Prove that sin⁡θ−cos⁡θ+1sin⁡θ+cos⁡θ−1=1sec⁡θ−tan⁡θ\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \dfrac{1}{\sec\theta - \tan\theta}, using the identity sec⁡2θ=1+tan⁡2θ\sec^2\theta = 1 + \tan^2\theta.

Exercise 8.3

Practice · 14
  1. Ex 8.3 Q1
    Express the trigonometric ratios sin⁡A\sin \text{A}, sec⁡A\sec \text{A} and tan⁡A\tan \text{A} in terms of cot⁡A\cot \text{A}.
  2. Ex 8.3 Q2
    Write all the other trigonometric ratios of ∠\angleA in terms of sec⁡A\sec \text{A}.
  3. Choose the correct option. Justify your choice.
    Ex 8.3 Q3 (i)
    9sec⁡2A−9tan⁡2A=9 \sec^2 \text{A} - 9 \tan^2 \text{A} =
    1. A.
      11
    2. B.
      99
    3. C.
      88
    4. D.
      00
  4. Ex 8.3 Q3 (ii)
    (1+tan⁡θ+sec⁡θ)(1+cot⁡θ−cosec⁡θ)=(1 + \tan\theta + \sec\theta)(1 + \cot\theta - \operatorname{cosec}\theta) =
    1. A.
      00
    2. B.
      11
    3. C.
      22
    4. D.
      −1-1
  5. Ex 8.3 Q3 (iii)
    (sec⁡A+tan⁡A)(1−sin⁡A)=(\sec \text{A} + \tan \text{A})(1 - \sin \text{A}) =
    1. A.
      sec⁡A\sec \text{A}
    2. B.
      sin⁡A\sin \text{A}
    3. C.
      cosec⁡A\operatorname{cosec} \text{A}
    4. D.
      cos⁡A\cos \text{A}
  6. Ex 8.3 Q3 (iv)
    1+tan⁡2A1+cot⁡2A=\dfrac{1 + \tan^2 \text{A}}{1 + \cot^2 \text{A}} =
    1. A.
      sec⁡2A\sec^2 \text{A}
    2. B.
      −1-1
    3. C.
      cot⁡2A\cot^2 \text{A}
    4. D.
      tan⁡2A\tan^2 \text{A}
  7. Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
    Ex 8.3 Q4 (i)
    (cosec⁡θ−cot⁡θ)2=1−cos⁡θ1+cos⁡θ(\operatorname{cosec}\theta - \cot\theta)^2 = \dfrac{1 - \cos\theta}{1 + \cos\theta}
  8. Ex 8.3 Q4 (ii)
    cos⁡A1+sin⁡A+1+sin⁡Acos⁡A=2sec⁡A\dfrac{\cos \text{A}}{1 + \sin \text{A}} + \dfrac{1 + \sin \text{A}}{\cos \text{A}} = 2 \sec \text{A}
  9. Ex 8.3 Q4 (iii)
    tan⁡θ1−cot⁡θ+cot⁡θ1−tan⁡θ=1+sec⁡θcosec⁡θ\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \sec\theta \operatorname{cosec}\theta [Hint : Write the expression in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta]
  10. Ex 8.3 Q4 (iv)
    1+sec⁡Asec⁡A=sin⁡2A1−cos⁡A\dfrac{1 + \sec \text{A}}{\sec \text{A}} = \dfrac{\sin^2 \text{A}}{1 - \cos \text{A}} [Hint : Simplify LHS and RHS separately]
  11. Ex 8.3 Q4 (v)
    cos⁡A−sin⁡A+1cos⁡A+sin⁡A−1=cosec⁡A+cot⁡A\dfrac{\cos \text{A} - \sin \text{A} + 1}{\cos \text{A} + \sin \text{A} - 1} = \operatorname{cosec} \text{A} + \cot \text{A}, using the identity cosec⁡2A=1+cot⁡2A\operatorname{cosec}^2 \text{A} = 1 + \cot^2 \text{A}.
  12. Ex 8.3 Q4 (vi)
    1+sin⁡A1−sin⁡A=sec⁡A+tan⁡A\sqrt{\dfrac{1 + \sin \text{A}}{1 - \sin \text{A}}} = \sec \text{A} + \tan \text{A}
  13. Ex 8.3 Q4 (vii)
    sin⁡θ−2sin⁡3θ2cos⁡3θ−cos⁡θ=tan⁡θ\dfrac{\sin\theta - 2\sin^3\theta}{2\cos^3\theta - \cos\theta} = \tan\theta
  14. Ex 8.3 Q4 (viii)
    (sin⁡A+cosec⁡A)2+(cos⁡A+sec⁡A)2=7+tan⁡2A+cot⁡2A(\sin \text{A} + \operatorname{cosec} \text{A})^2 + (\cos \text{A} + \sec \text{A})^2 = 7 + \tan^2 \text{A} + \cot^2 \text{A}