Relationships Between Ratios MCQs Quiz | Class 10

This quiz covers important relationships between trigonometric ratios for Class X Mathematics, Unit V: Trigonometry. It focuses on reciprocal identities like sin=1/cosec, cos=1/sec, tan=1/cot, and also tan=sin/cos and cot=cos/sin. Test your understanding with 10 multiple-choice questions. Submit your answers and then download a detailed PDF of your results for review.

Understanding Relationships Between Trigonometric Ratios

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. In Class 10, a foundational understanding of trigonometric ratios and their inter-relationships is crucial. This section will help you revise the key concepts of reciprocal identities and quotient identities.

Reciprocal Identities

The six trigonometric ratios—sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot)—are related to each other through reciprocal identities. A reciprocal identity means that one ratio is the inverse of another.

  • Sine and Cosecant: The cosecant of an angle is the reciprocal of its sine.
    • sin A = 1 / cosec A
    • cosec A = 1 / sin A
    • sin A * cosec A = 1
  • Cosine and Secant: The secant of an angle is the reciprocal of its cosine.
    • cos A = 1 / sec A
    • sec A = 1 / cos A
    • cos A * sec A = 1
  • Tangent and Cotangent: The cotangent of an angle is the reciprocal of its tangent.
    • tan A = 1 / cot A
    • cot A = 1 / tan A
    • tan A * cot A = 1

Quotient Identities

The tangent and cotangent ratios can also be expressed in terms of sine and cosine. These are known as quotient identities.

  • Tangent: The tangent of an angle is the ratio of its sine to its cosine.
    • tan A = sin A / cos A
  • Cotangent: The cotangent of an angle is the ratio of its cosine to its sine.
    • cot A = cos A / sin A

Summary Table of Identities

Identity Type Relationship
Reciprocal sin A = 1 / cosec A
Reciprocal cos A = 1 / sec A
Reciprocal tan A = 1 / cot A
Quotient tan A = sin A / cos A
Quotient cot A = cos A / sin A

Quick Revision Checklist

  • Remember that “SOH CAH TOA” helps define the basic ratios for a right-angled triangle.
  • Cosecant is always 1 divided by Sine.
  • Secant is always 1 divided by Cosine.
  • Cotangent is always 1 divided by Tangent.
  • Tangent can be expressed as Sine divided by Cosine.
  • Cotangent can be expressed as Cosine divided by Sine.
  • These identities are fundamental for solving more complex trigonometric problems.

Practice Questions

Test your understanding with these additional practice questions:

  1. If sin A = 0.5, what is cosec A?
  2. Express sec A in terms of cos A.
  3. If tan A = 1, find the value of sin A / cos A.
  4. Simplify: (sin A / cos A) * cot A.
  5. If cosec A = 2, calculate sin A.

Keep practicing these fundamental relationships to build a strong foundation in trigonometry!