Long Answer Questions: Introduction to Trigonometry

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

Q1: In the adjoining figure, ABCD is a rectanlge with breadth BC = 7 cm and ∠CAB = 30°. Find the length of side AB of the rectangle and length of diagonal AC. If the ∠CAB = 60°, then what is the size of the side AB of the rectangle. [Use √3 = 1.73 and √2 = 1.41, if required)
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Q2: Prove that:

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Q3: If x = r sin A cos C, y = r sin A sin C and z = r cos A, then prove that x2 + y2 + z2 = r2.
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x = r sin A cos C; y = r sin A sin C; z = r cos A
L.H.S. x2 + y2 + z2 = 2 sin2 A cos2C + r2 sin2 A sin2 C + r2 cos2 A
= r2 sin2 A(cos2 C + sin2 C) + r2 cos2 A
= r2 sin2 A + r2 cos2 A … [cos2θ + sin2θ = 1
= r2 (sin2 A + cos2 A) = r2 = R.H.S.

Q4: Prove that: (cot A + sec B)2 – (tan B – cosec A)2 = 2(cot A . sec B + tan B. cosec A)
Ans:
L.H.S.
= (cot A + sec B)2 – (tan B – cosec A)2
= cot2 A + sec2 B + 2 cot A sec B – (tan2 B + cosec2 A – 2 tan B cosec A)
= cot2 A + sec2 B + 2 cot A sec B – tan2 B – cosec2 A + 2 tan B cosec A
= (sec2 B – tan2 B) – (cosec2 A – cot2 A) + 2(cot A sec B + tan B cosec A)
= 1 – 1 + 2(cot A sec B + tan B cosec A) … [∵ sec2B – tan2 B = 1
cosec2A – cot2 A = 1
= 2(cot A . sec B + tan B . cosec A) = R.H.S.

Q5: Prove the following trigonometric identities: sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A.
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L.H.S. = sin A (1 + tan A) + cos A (1 + cot A)

Q6: Prove the identity:

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Q7: Prove that: (1 + cot A + tan A). (sin A – cos A) =
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Q8: Prove that:

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Q9: Find the value of:

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Q10: Prove that:  = 2 cosec A
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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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## Mathematics (Maths) Class 10

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## FAQs on Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

 1. What is trigonometry and why is it important in Class 10 mathematics?
Ans. Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is important in Class 10 mathematics as it helps in solving various real-life problems involving measurements, distances, heights, and angles.
 2. How do we find the values of trigonometric ratios in Class 10?
Ans. In Class 10, we can find the values of trigonometric ratios using tables, calculators, or by using special triangles (like 30-60-90 and 45-45-90 triangles) to determine the ratios. We can also use the Pythagorean theorem and the definitions of sine, cosine, and tangent to find the values.
 3. What are the basic trigonometric ratios and how are they used in Class 10?
Ans. The basic trigonometric ratios are sine, cosine, and tangent. In Class 10, these ratios are used to find the relationships between the angles and sides of a right triangle. For example, the sine ratio (sinθ) is defined as the ratio of the length of the side opposite the angle θ to the hypotenuse, cosine ratio (cosθ) is the ratio of the adjacent side to the hypotenuse, and tangent ratio (tanθ) is the ratio of the opposite side to the adjacent side.
 4. How can trigonometry be applied in real-life situations?
Ans. Trigonometry can be applied in various real-life situations, such as measuring heights and distances, calculating the angles of elevation and depression, analyzing the motion of objects, designing structures, and solving navigation problems. It is also used in fields like engineering, physics, architecture, and astronomy.
 5. What are the practical applications of trigonometry in Class 10 mathematics?
Ans. In Class 10 mathematics, trigonometry is used to solve problems related to heights and distances, angles of elevation and depression, and the concept of bearings. It helps in determining the heights of buildings or mountains, finding the distances between two points (without actually measuring them), and calculating the angles of inclination or depression.

## Mathematics (Maths) Class 10

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