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 Trigonometry

Class 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 TrigonometryClass 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 are the basic trigonometric ratios?
Ans. The basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). They are defined as follows for a right triangle: - sin(θ) = opposite side / hypotenuse - cos(θ) = adjacent side / hypotenuse - tan(θ) = opposite side / adjacent side.
2. How do you remember the trigonometric ratios?
Ans. A common mnemonic to remember the basic trigonometric ratios is "SOH CAH TOA," where: - SOH stands for Sine = Opposite / Hypotenuse - CAH stands for Cosine = Adjacent / Hypotenuse - TOA stands for Tangent = Opposite / Adjacent.
3. What is the unit circle and its significance in trigonometry?
Ans. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is significant in trigonometry because it provides a way to define the sine and cosine functions for all angles, not just those in right triangles. The coordinates of points on the circle correspond to the cosine and sine of the angle formed with the positive x-axis.
4. How do you convert degrees to radians in trigonometry?
Ans. To convert degrees to radians, you can use the formula: Radians = Degrees × (π / 180). For example, to convert 90 degrees to radians, you would calculate 90 × (π / 180) = π/2 radians.
5. What are the Pythagorean identities in trigonometry?
Ans. The Pythagorean identities relate the squares of the sine, cosine, and tangent functions. The primary identity is: sin²(θ) + cos²(θ) = 1. Other identities derived from this include: 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ).
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