# NCERT Solutions for Class 10 Maths Unit 8

## Introduction to Trigonometry Class 10

### Unit 8 Introduction to Trigonometry Exercise 8.1, 8.2, 8.3, 8.4 Solutions

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined.

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA.

### Exercise 8.1 : Solutions of Questions on Page Number : 181

Q1 :

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine

(i) sin A, cos A

(ii) sin C, cos C

Applying Pythagoras theorem for ΔABC, we obtain

AC2 = AB2 + BC2

= (24 cm)2 + (7 cm)2

= (576 + 49) cm2

= 625 cm2

∴ AC = cm = 25 cm

(i) sin A =

cos A =

(ii)

sin C =

cos C =

Q2 :

In the given figure find tan P - cot R

Applying Pythagoras theorem for ΔPQR, we obtain

PR2 = PQ2 + QR2

(13 cm)2 = (12 cm)2 + QR2

169 cm2 = 144 cm2 + QR2

25 cm2 = QR2

QR = 5 cm

tan P - cot R =

Q3 :

If sin A =, calculate cos A and tan A.

Q4 :

Given 15 cot A = 8. Find sin A and sec A

Q5 :

Given sec θ =, calculate all other trigonometric ratios.

Q6 :

If ∠ A and ∠ B are acute angles such that cos A = cos B, then show that

∠ A = ∠ B.

Q7 :

If cot θ =, evaluate

(i) (ii) cot2 θ

Q8 :

If 3 cot A = 4, Check whether

Q9 :

In ΔABC, right angled at B. If, find the value of

(i) sin A cos C + cos A sin C

(ii) cos A cos C - sin A sin C

Q10 :

In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

Q11 :

(i) The value of tan A is always less than 1.

(ii) sec A =for some value of angle A.

(iii) cos A is the abbreviation used for the cosecant of angle A.

(iv) cot A is the product of cot and A

(v) sin θ =, for some angle θ

### Exercise 8.2 : Solutions of Questions on Page Number : 187

Q1 :

Evaluate the following

(i) sin60° cos30° + sin30° cos 60°

(ii) 2tan245° + cos230° - sin260°

(iii)

(iv)

(v)

Q2 :

Choose the correct option and justify your choice.

(i)

(A). sin60°

(B). cos60°

(C). tan60°

(D). sin30°

(ii)

(A). tan90°

(B). 1

(C). sin45°

(D). 0

(iii) sin2A = 2sinA is true when A =

(A). 0°

(B). 30°

(C). 45°

(D). 60°

(iv)

(A). cos60°

(B). sin60°

(C). tan60°

(D). sin30°

Q3 :

If and;

0° < A + B 90°, A > B find A and B.

Q4 :

(i) sin(A + B) = sin A + sin B

(ii) The value of sinÃŽÂ¸increases as ÃŽÂ¸increases

(iii) The value of cos ÃŽÂ¸increases as ÃŽÂ¸increases

(iv) sinÃŽÂ¸= cos ÃŽÂ¸ for all values of ÃŽÂ¸

(v) cot A is not defined for A = 0°

### Exercise 8.3 : Solutions of Questions on Page Number : 189

Q1 :

Evaluate

(I)

(II)

(III) cos 48° - sin 42°

(IV)cosec 31° - sec 59°

Q2 :

Show that

(I) tan 48° tan 23° tan 42° tan 67° = 1

(II)cos 38° cos 52° - sin 38° sin 52° = 0

Q3 :

If tan 2A = cot (A- 18°), where 2A is an acute angle, find the value of A.

Q4 :

If tan A = cot B, prove that A + B = 90°

Q5 :

If sec 4A = cosec (A- 20°), where 4A is an acute angle, find the value of A.

Q6 :

If A, Band C are interior angles of a triangle ABC then show that

Q7 :

Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

### Exercise 8.4 : Solutions of Questions on Page Number : 193

Q1 :

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Q2 :

Write all the other trigonometric ratios of ∠ A in terms of sec A.

Q3 :

Evaluate

(i)

(ii) sin25° cos65° + cos25° sin65°

Q4 :

Choose the correct option. Justify your choice.

(i) 9 sec2 A - 9 tan2 A =

(A) 1

(B) 9

(C) 8

(D) 0

(ii) (1 + tan θ + sec θ) (1 + cot θ - cosec θ)

(A) 0

(B) 1

(C) 2

(D) - 1

(iii) (secA + tanA) (1 - sinA) =

(A) secA

(B) sinA

(C) cosecA

(D) cosA

(iv)

(A) sec2 A

(B) - 1

(C) cot2 A

(D) tan2 A

Q5 :

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.