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Introduction to Trigonometry — Class 10 solved problems

37 problems from this chapter, each solved step by step.

  1. Evaluate: sin 30 degrees times cos 60 degrees + cos 30 degrees times sin 60 degrees.

    1

  2. If sin(A) = 3/5 and cos(B) = 12/13, find sin(A+B) and cos(A-B).

    sin(A+B) = (56)/(65) and cos(A-B) = (63)/(65)

  3. Prove that (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ.

    (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ

  4. Trigonometry If sin θ = (5)/(13) and θ is acute, find: * cos θ * tan θ

    cos θ = (12)/(13), tan θ = (5)/(12)

  5. Prove that (1 + tan² A) / (1 + cot² A) = tan² A.

    (1 + tan² A)/(1 + cot² A) = tan² A

  6. Prove the identity: (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ) = 2 cosec θ.

    The identity (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ) = 2 cosec θ is proven.

  7. If tan θ =1 and θ is acute find the value of sin θ + cosθ

    The value of sin θ + cos θ is √2.

  8. Find the value of sin 30 degrees + cos 60 degrees

    The value of sin 30^(°) + cos 60^(°) is 1.

  9. In a right-angled triangle △ ABC, the right angle is at B. If AB = 24 cm and BC = 7 cm, find the values of sin A, cos A, sin C, and cos C.

    sin A = (7)/(25), cos A = (24)/(25), sin C = (24)/(25), cos C = (7)/(25)

  10. TRIGONOMETRY (10th Class) Exercise 8.1 Ques 1: In ABC, right angle at B, AB = 24 cm, BC = 7 cm. Determined: (i) sin A, cos A (ii) sin C, cos C Sol: Let us draw

    (i) sin A = (7)/(25), cos A = (24)/(25) (ii) sin C = (24)/(25), cos C = (7)/(25)

  11. If cos(A+B) = 0 and sin(A-B) = (1)/(2), then find the value of A and B, where A and B are acute angles. (trigonometry, acute angles, sum and difference identiti

    A = 60^° and B = 30^°

  12. If 3‾√ cot2θ – 4 cot θ + 3‾√ = 0, then find the value of cot2 θ + tan2θ. (2013) Solution:

    The value of cot² θ + tan²θ is (10)/(3).

  13. (a) (i) Prove that: (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ (ii) Prove that: (sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = (2)/(2sin²

    The identities are proven as follows: (i) (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ (ii) (sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = (

  14. Prove that: (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = (1)/(sec θ - tan θ).

    (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = (1)/(sec θ - tan θ)

  15. (a) If cos (A + B) = (1)/(2) and tan (A - B) = (1)/(√3), where 0 ≤ A + B ≤ 90^°, then find the value of sec (2A - 3B). OR (b) Find the value of x such that, 3 t

    √2

  16. The value of (1 + tan² θ)/(1 + cot² θ) is:

    tan² θ

  17. If 5 tan θ = 2, then the value of (10 sin θ - 2 cos θ)/(5 sin θ + 3 cos θ) is:

    (2)/(5)

  18. cot² θ - (1)/(sin² θ) is equal to:

    -1

  19. (a) If cos θ = (3)/(5), find the value of (tan θ + sin θ)/(cosec θ - cot θ). OR (b) For θ = 30^°, verify that cos 2θ = cos² θ - sin² θ.

    (64)/(15)

  20. If cosec B = (√3)/(2) and A + B = 90^°, then the value of sec A is:

    The problem statement contains an invalid value for cosec B, as (√3)/(2) ≈ 0.866, which is less than 1. The cosecant of a real angle must be greater than or equ

  21. The maximum value of (1)/(cosec θ), (0 ≤ θ ≤ 90^°) is:

    1

  22. In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.

    Since 2 sin A cos A = 1, the statement is verified.

  23. Given tan A = (4)/(3), find the other trigonometric ratios of the angle A.

    The other trigonometric ratios are sin A = (4)/(5), cos A = (3)/(5), cosec A = (5)/(4), sec A = (5)/(3), and cot A = (3)/(4).

  24. Consider ACB, right-angled at C, in which AB = 29 units, BC = 21 units and ABC = θ. Determine the values of (i) cos² θ + sin² θ, (ii) cos² θ - sin² θ.

    (i) cos² θ + sin² θ = 1 (ii) cos² θ - sin² θ = (41)/(841)

  25. If B and Q are acute angles such that sin B = sin Q, then prove that B = Q.

    B = Q

  26. If sin A = 3/5 and A is an acute angle, find the values of cos A and tan A.

    cos A = (4)/(5), tan A = (3)/(4)

  27. In triangle ABC, C=90^°. Point D is the midpoint of BC, and AD=BC. Find sin BAD.

    The value of sin BAD is (√21)/(14).

  28. In triangle ABC, BAC=75^°, ACB=60^°, and AB=8√2. Find BC.

    BC = 8 + (8√3)/(3)

  29. ACB लीजिए जिसका कोण C समकोण है जिसमें AB = 29 इकाई, BC = 21 इकाई और ABC = θ हैं तो निम्नलिखित के मान ज्ञात कीजिए: (i) cos² θ + sin² θ, (ii) cos² θ - sin² θ।

    (i) cos² θ + sin² θ = 1 और (ii) cos² θ - sin² θ = (41)/(841)

  30. In a right-angled triangle ABC, right-angled at B, if AB = 3 cm and BC = 4 cm, find sin A and cos A.

    sin A = (4)/(5) and cos A = (3)/(5)

  31. एक समकोण त्रिभुज ABC में, जिसका कोण B समकोण है, यदि tan A = 1 तो सत्यापित कीजिए कि 2 sin A cos A = 1।

    सत्यापित हो गया कि 2 sin A cos A = 1।

  32. The value of ( sin 70^°) ( cot 10^° cot 70^° - 1) is

    1

  33. In a square grid, the position of ABC is shown. Find tan B.

    tan B = 2

  34. In the square grid shown, what is tan AOB?

    (15)/(8)

  35. In OPQ, right-angled at P, OP = 7 cm and OQ - PQ = 1 cm. Determine the values of sin Q and cos Q.

    The values are sin Q = (7)/(25) and cos Q = (24)/(25).

  36. यदि B और Q ऐसे न्यूनकोण हों जिससे कि sin B = sin Q, तो सिद्ध कीजिए कि B = Q।

    चूंकि sin B = sin Q दिया गया है, और हमने दिखाया कि यह दो समरूप त्रिभुजों को दर्शाता है (जहाँ B और Q संगत कोण हैं), इसलिए B = Q।

  37. OPQ में, जिसका कोण P समकोण है, OP = 7 cm और OQ - PQ = 1 cm, sin Q और cos Q के मान ज्ञात कीजिए।

    sin Q = (7)/(25) और cos Q = (24)/(25)

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