Mathematics · Textbook solutions

Trigonometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 37 questions

Practice Set 8.1

14 q
  1. In the Fig. 8.12, ∠R\angle R is the right angle of ΔPQR\Delta PQR. Write the following ratios. Fig. 8.12 shows ΔPQR\Delta PQR with the right-angle square marked at RR — PP at the left, RR at the top right and QQ at the bottom right — so PQPQ is the hypotenuse while PRPR and RQRQ are the two legs. No side lengths are printed; write each ratio using the side names.
    Ex 8.1 Q1(i)
    sin⁡P\sin P
  2. Ex 8.1 Q1(ii)
    cos⁡Q\cos Q
  3. Ex 8.1 Q1(iii)
    tan⁡P\tan P
  4. Ex 8.1 Q1(iv)
    tan⁡Q\tan Q
  5. In the right angled ΔXYZ\Delta XYZ, ∠XYZ=90∘\angle XYZ = 90^\circ and a,b,ca, b, c are the lengths of the sides as shown in the figure. Write the following ratios. In Fig. 8.13 the right-angle square is at YY, with YY at the top left, XX at the top right and ZZ at the bottom; the side labels are a=YZa = YZ, b=XYb = XY and c=XZc = XZ, so cc is the hypotenuse and aa, bb are the two legs.
    Ex 8.1 Q2(i)
    sin⁡X\sin X
  6. Ex 8.1 Q2(ii)
    tan⁡Z\tan Z
  7. Ex 8.1 Q2(iii)
    cos⁡X\cos X
  8. Ex 8.1 Q2(iv)
    tan⁡X\tan X
  9. In right angled ΔLMN\Delta LMN, ∠LMN=90∘\angle LMN = 90^\circ, ∠L=50∘\angle L = 50^\circ and ∠N=40∘\angle N = 40^\circ, write the following ratios. In Fig. 8.14 the right-angle square is at MM, with LL at the top, MM at the bottom left and NN at the bottom right; the 50∘50^\circ is marked at LL and the 40∘40^\circ at NN, so LNLN is the hypotenuse while LMLM and MNMN are the two legs. No side lengths are printed; write each ratio using the side names.
    Ex 8.1 Q3(i)
    sin⁡50∘\sin 50^\circ
  10. Ex 8.1 Q3(ii)
    cos⁡50∘\cos 50^\circ
  11. Ex 8.1 Q3(iii)
    tan⁡40∘\tan 40^\circ
  12. Ex 8.1 Q3(iv)
    cos⁡40∘\cos 40^\circ
  13. In the figure 8.15, ∠PQR=90∘\angle PQR = 90^\circ, ∠PQS=90∘\angle PQS = 90^\circ, ∠PRQ=α\angle PRQ = \alpha and ∠QPS=θ\angle QPS = \theta. Write the following trigonometric ratios. In Fig. 8.15 the points RR, QQ, SS lie in that order on one straight line and PP is above QQ, with the right-angle square marked at QQ so that PQ⊥RSPQ \perp RS; α\alpha is the angle at RR between RPRP and RQRQ, and θ\theta is the angle at PP between PQPQ and PSPS. Thus ΔPQR\Delta PQR is right angled at QQ with hypotenuse PRPR, and ΔPQS\Delta PQS is right angled at QQ with hypotenuse PSPS. No side lengths are printed; write each ratio using the side names.
    Ex 8.1 Q4(i)
    sin⁡α\sin \alpha, cos⁡α\cos \alpha, tan⁡α\tan \alpha
  14. Ex 8.1 Q4(ii)
    sin⁡θ\sin \theta, cos⁡θ\cos \theta, tan⁡θ\tan \theta

Ratios of Particular Angles

4 q

Solved Examples

Worked · 4
  1. 8.2 SolvedEx.1
    Find the value of 2tan⁡45∘+cos⁡30∘−sin⁡60∘2\tan 45^\circ + \cos 30^\circ - \sin 60^\circ
  2. 8.2 SolvedEx.2
    Find the value of cos⁡56∘sin⁡34∘\dfrac{\cos 56^\circ}{\sin 34^\circ}
  3. 8.2 SolvedEx.3
    In right angled Δ\Delta ACB, if ∠C=90∘\angle C = 90^\circ, AC=3AC = 3, BC=4BC = 4. Find the ratios sin⁡A\sin A, sin⁡B\sin B, cos⁡A\cos A, tan⁡B\tan B.
  4. 8.2 SolvedEx.4
    In right angled triangle Δ\Delta PQR, ∠Q=90∘\angle Q = 90^\circ, ∠R=θ\angle R = \theta and if sin⁡θ=513\sin \theta = \dfrac{5}{13} then find cos⁡θ\cos \theta and tan⁡θ\tan \theta.

Practice Set 8.2

9 q
  1. Ex 8.2 Q1
    In the following table, a ratio is given in each column. Find the remaining two ratios in the column and complete the table.
    123456789
    sin⁡θ\sin \theta1161\frac{11}{61}12\frac{1}{2}35\frac{3}{5}
    cos⁡θ\cos \theta3537\frac{35}{37}13\frac{1}{\sqrt{3}}
    tan⁡θ\tan \theta112120\frac{21}{20}815\frac{8}{15}122\frac{1}{2\sqrt{2}}
    (The book's table prints no header row; the nine columns are numbered 1 to 9 here for reference. Each column carries exactly one given ratio and two empty cells to be filled.)
  2. Find the values of -
    Ex 8.2 Q2(i)
    5sin⁡30∘+3tan⁡45∘5 \sin 30^{\circ} + 3 \tan 45^{\circ}
  3. Ex 8.2 Q2(ii)
    45tan⁡260∘+3sin⁡260∘\frac{4}{5} \tan^{2} 60^{\circ} + 3 \sin^{2} 60^{\circ}
  4. Ex 8.2 Q2(iii)
    2sin⁡30∘+cos⁡0∘+3sin⁡90∘2 \sin 30^{\circ} + \cos 0^{\circ} + 3 \sin 90^{\circ}
  5. Ex 8.2 Q2(iv)
    tan⁡60sin⁡60+cos⁡60\frac{\tan 60}{\sin 60 + \cos 60}
  6. Ex 8.2 Q2(v)
    cos⁡245∘+sin⁡230∘\cos^{2} 45^{\circ} + \sin^{2} 30^{\circ}
  7. Ex 8.2 Q2(vi)
    cos⁡60∘×cos⁡30∘+sin⁡60∘×sin⁡30∘\cos 60^{\circ} \times \cos 30^{\circ} + \sin 60^{\circ} \times \sin 30^{\circ}
  8. Ex 8.2 Q3
    If sin⁡θ=45\sin \theta = \frac{4}{5} then find cos⁡θ\cos \theta
  9. Ex 8.2 Q4
    If cos⁡θ=1517\cos \theta = \frac{15}{17} then find sin⁡θ\sin \theta

Problem Set 8

10 q
  1. Choose the correct alternative answer for following multiple choice questions.
    Prob Q1(i)
    Which of the following statements is true ?
    1. A.
      sin⁡θ=cos⁡(90−θ)\sin \theta = \cos (90 - \theta)
    2. B.
      cos⁡θ=tan⁡(90−θ)\cos \theta = \tan (90 - \theta)
    3. C.
      sin⁡θ=tan⁡(90−θ)\sin \theta = \tan (90 - \theta)
    4. D.
      tan⁡θ=tan⁡(90−θ)\tan \theta = \tan (90 - \theta)
  2. Prob Q1(ii)
    Which of the following is the value of sin⁡90∘\sin 90^{\circ} ?
    1. A.
      32\frac{\sqrt{3}}{2}
    2. B.
      00
    3. C.
      12\frac{1}{2}
    4. D.
      11
  3. Prob Q1(iii)
    2tan⁡45∘+cos⁡45∘−sin⁡45∘=2 \tan 45^{\circ} + \cos 45^{\circ} - \sin 45^{\circ} = ?
    1. A.
      00
    2. B.
      11
    3. C.
      22
    4. D.
      33
  4. Prob Q1(iv)
    cos⁡28∘sin⁡62∘=\frac{\cos 28^{\circ}}{\sin 62^{\circ}} = ?
    1. A.
      22
    2. B.
      −1-1
    3. C.
      00
    4. D.
      11
  5. Prob Q2
    In right angled △\triangle TSU, TS = 5, ∠\angleS = 90∘90^{\circ}, SU = 12 then find sin T, cos T, tan T. Similarly find sin U, cos U, tan U.
  6. Prob Q3
    In right angled △\triangle YXZ, ∠\angleX = 90∘90^{\circ}, XZ = 8 cm, YZ = 17 cm, find sin Y, cos Y, tan Y, sin Z, cos Z, tan Z.
  7. Prob Q4
    In right angled △\triangle LMN, if ∠\angleN = θ\theta, ∠\angleM = 90∘90^{\circ}, cos⁡θ=2425\cos \theta = \frac{24}{25}, find sin⁡θ\sin \theta and tan⁡θ\tan \theta Similarly, find (sin⁡2θ)(\sin^{2} \theta) and (cos⁡2θ)(\cos^{2} \theta).
  8. Fill in the blanks.
    Prob Q5(i)
    sin⁡20∘=cos⁡\sin 20^{\circ} = \cos ____∘^{\circ}
  9. Prob Q5(ii)
    tan⁡30∘×tan⁡\tan 30^{\circ} \times \tan ____∘=1^{\circ} = 1
  10. Prob Q5(iii)
    cos⁡40∘=sin⁡\cos 40^{\circ} = \sin ____∘^{\circ}