Class 10 Exam  >  Class 10 Notes  >  Mathematics (Maths) Class 10  >  Short Answer Questions: Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q1: It is given that tan (θ1 + θ2) =  Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry where θ1 and θ2 are acute angles.
Calculate θ1 + θ2 when tan θ1Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Now, tan (θ1 + θ2) = 1 ⇒ θ1 + θ2 = 45°.
Q2: Prove that:   Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
using tanθ=sinθ/cosθ  and cos2θ=1-sin2θ   as sin2θ+cos2θ=1

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q3: Prove that:  (sin4 θ – cos4 θ +1) cosec2 θ = 2  
Sol:  L.H.S.
= (sin4 θ – cos4 θ + 1) cosec2 θ
= [(sin2 θ)2 – (cos2 θ)2 + 1] cosec2 θ             as    [a2-b2=(a-b)(a+b)]
= [(sin2 θ – cos2 θ) (sin2 θ + cos2 θ) + 1] cosec2 θ        as   [ sin2 θ + cos2 θ = 1]
= [(sin2 θ – cos2 θ) *1 + 1] cosec2 θ
= [sin2 θ – cos2 θ+1]  cosec2 θ
= [(sin2 θ + (1 –cos2 θ)] cosec2 θ   [ 1 – cos2 θ = sin2 θ]
= [sin2 θ + sin2 θ] cosec2 θ  
= 2 sin2 θ . cosec2 θ
= 2 = RHS [∵ sin θ . cosec θ = 1]
Q4: Prove that: sec2 θ + cosec2 θ = sec2 θ · cosec2 θ 

Sol: L.H.S. = sec2 θ + cosec2 θ

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q5: Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q6:  Given that α + β = 90°, show that:  Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: ∵ α + β =90°
∵ β = (90 – a)
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Q7:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

using a3+b3=(a+b)(a2+b2-ab)   and a3-b3=(a-b)(a2+b2+ab) in numerator of these terms

and also sin2θ +cos2θ =1

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q8:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

using tanθ=sinθ/cosθ  and then taking LCM

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q9: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q10:  Prove that: sin6θ + cos6θ + 3sin2θ cos2θ = 1.

Sol: ∵ sinθ + cos2θ = 1
∵ (sin2 θ + cos2 θ)3 = (1)3 = 1
⇒ (sin2 θ) 3 + (cos2 θ)3 + 3 sin2 θ . cos2 θ (sin2 θ + cos2 θ) = 1
⇒ sin6 θ + cos6 θ + 3 sin2 θ . cos2θ (1) = 1
⇒ sin6 θ + cos6 θ + 3 sin2 θ . cos2 θ = 1

Q11:  Prove that: a2 + b2 = x2 + y2 when a cos θ − b sin θ = x and a sin θ + b cos θ =y.

Sol:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q12:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol: 
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q13:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry 

Sol:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q14:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q15:

 Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: 
Class 10 Maths Chapter 8 Question Answers - Introduction to TrigonometryClass 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Q16: For an acute angle θ, show that: (sin θ − cosec θ) (cos θ − sec θ) Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q17: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: 
Class 10 Maths Chapter 8 Question Answers - Introduction to TrigonometryClass 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Q18:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q19: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q20: Without using trigonometric tables evaluate:
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q21: If tan (A + B) = √3 and tan (A − B) = 1, 0° < A + B < 90°; A > B, then find A and B.

Sol: We have: tan (A + B) = 3 (Given)
tan 60° = 3 (From the table)
⇒ A + B = 60° ...(1)
Also,tan (A − B)= 1 [Given]
and cosec 60° = 2 and cos 90° = 0
⇒ A − B = 45 ...(2)
Adding (1) and (2),
2A = 60° + 45° = 105°

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

From (2), 52.5° − B = 45°
⇒ B = 52.5° − 45° = 7.5°
Thus, A = 52.5° and B = 7.5°.

Q22: If tan (2A) = cot (A − 21°), where 2A is an acute angle, then find the value of A.

Sol: We have: tan (2A) = cot (A − 21°)
∵ cot (90°− θ) = tan θ
∴ cot (90°− 2A) = tan 2A
⇒ cot (90°− 2A) = cot (A − 21)°
⇒ 90 − 2A = A − 21°
⇒ − 2A − A = − 21°− 90°
⇒ − 3A = − 111°

⇒  Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q23: If sin 3A = cos (A − 10°), then find the value of A, where 3A is an acute angle.

Sol: We have:
sin 3A = cos (A − 10°)
∵ cos (90° − θ) = sin θ
∴ cos (90° − 3A) = sin 3A
⇒ 90° − 3A = A − 10°
⇒ −3A − A = −10° − 90°
⇒ −4A = −100°

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q24: If sec 2A = cosec (A − 27°), then find the value of A, where 2A is an acute angle.

Sol: We have:

sec 2A = cosec (A − 27°) ...(1)
∵ sec θ = cosec (90° − θ)
∴ sec 2A = cosec (90° − 2A) ...(2)

From (1) and (2), we get
A − 27° = 90° − 2A
⇒ A + 2A = 90 + 27° = 117°
⇒ 3A = 117°
⇒  Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q25: Simplify:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry + sin θ cos θ

Sol: We have:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q26:

 Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q27: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol: 

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Class 10 Maths Chapter 8 Question Answers - Introduction to TrigonometryClass 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

= 1 = R.H.S.

Q28: Without using trigonometrical tables, evaluate:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Sol:

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to TrigonometryClass 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

[∵ sin (90° − θ) = cos θ, cos (90° − θ) = sin θ, cosec (90° − θ) = sec θ, and tan (90° − θ) = cot θ]
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry
Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q29: Using Geometry, find the value of sin 60°.

Sol: Let us consider an equilateral ΔABC and draw AD ⊥ BC.
Since, each angle of an equilateral triangle = 60°
∴∠A = ∠B = ∠C = 60°

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Let AB = BC = AC = 2a

In ΔABD and ΔACD, we have:
AB = AC   [Given]
∠ADB = ∠ADC = 90°    [Construction]
AD = AD [Construction]
⇒ ΔABD ≅ ΔACD
⇒ BD = CD

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Now, using Pythagoras theorem, in right ΔABD,

AD2 = AB2 − BD2
= (2a)2 − a2
= 4a2 − a2
= 3a2

Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

∴  Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Thus,  Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

The document Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

1. What is trigonometry?
Ans. Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of trigonometric functions such as sine, cosine, and tangent, and their applications in solving various problems related to angles and distances.
2. How is trigonometry useful in real life?
Ans. Trigonometry has various real-life applications, such as in architecture, engineering, navigation, and physics. It is used to calculate distances, heights, angles, and trajectories in these fields. For example, trigonometry helps engineers design and construct buildings, bridges, and roads.
3. What are the basic trigonometric ratios?
Ans. The basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios represent the relationships between the sides of a right triangle. Sine is the ratio of the length of the side opposite the angle to the hypotenuse, cosine is the ratio of the length of the adjacent side to the hypotenuse, and tangent is the ratio of the length of the opposite side to the adjacent side.
4. How do you find the value of trigonometric functions?
Ans. The value of trigonometric functions can be found using a calculator or trigonometric tables. For example, to find the sine of an angle, divide the length of the side opposite the angle by the length of the hypotenuse. Similarly, to find the cosine, divide the length of the adjacent side by the length of the hypotenuse. The tangent can be found by dividing the length of the opposite side by the length of the adjacent side.
5. What are the applications of trigonometry in solving triangles?
Ans. Trigonometry is used to solve triangles by finding the unknown sides or angles. It is particularly useful in solving right triangles, where one angle is 90 degrees. By using the trigonometric ratios and the known values of sides or angles, we can determine the missing values. This is helpful in various fields such as surveying, navigation, and physics.
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