Standard Values: 30°, 45°, 60° MCQs Quiz | Class 10

This Class X Mathematics (Code 041) quiz from Unit V: Trigonometry focuses on Standard Values: 30°, 45°, 60° MCQs. Test your understanding of the values of all trigonometric ratios at these specific angles. Submit your answers to see your score and download a detailed answer PDF.

Understanding Standard Trigonometric Values (30°, 45°, 60°)

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. In particular, we focus on right-angled triangles. Certain angles, like 30 degrees, 45 degrees, and 60 degrees, are considered “standard” because their trigonometric ratio values (sine, cosine, tangent, cosecant, secant, cotangent) can be easily derived and are frequently used in problems.

Key Concepts

  • Trigonometric Ratios: For a right-angled triangle, with respect to an acute angle:
    • Sine (sin) = Opposite / Hypotenuse
    • Cosine (cos) = Adjacent / Hypotenuse
    • Tangent (tan) = Opposite / Adjacent
    • Cosecant (cosec) = 1 / sin = Hypotenuse / Opposite
    • Secant (sec) = 1 / cos = Hypotenuse / Adjacent
    • Cotangent (cot) = 1 / tan = Adjacent / Opposite
  • Derivation for 45°: Consider an isosceles right-angled triangle with two equal sides of length ‘a’. By Pythagoras theorem, the hypotenuse would be a*sqrt(2). From this, sin 45° = a / (a*sqrt(2)) = 1/sqrt(2), and so on.
  • Derivation for 30° and 60°: Consider an equilateral triangle with side length ‘2a’. Draw an altitude from one vertex to the opposite side. This altitude bisects the opposite side and the angle at the vertex, forming two right-angled triangles. In one such triangle, angles are 30°, 60°, and 90°. The sides would be a, a*sqrt(3), and 2a. From this, sin 30° = a / (2a) = 1/2, and sin 60° = (a*sqrt(3)) / (2a) = sqrt(3)/2, and so forth.

Table of Standard Trigonometric Values

It is crucial to memorize these values for quick problem-solving.

Ratio / Angle 0° 30° 45° 60° 90°
sin A 0 1/2 1/sqrt(2) sqrt(3)/2 1
cos A 1 sqrt(3)/2 1/sqrt(2) 1/2 0
tan A 0 1/sqrt(3) 1 sqrt(3) Undefined
cosec A Undefined 2 sqrt(2) 2/sqrt(3) 1
sec A 1 2/sqrt(3) sqrt(2) 2 Undefined
cot A Undefined sqrt(3) 1 1/sqrt(3) 0

Tips for Remembering

  • Finger Trick: There’s a popular finger trick for sin and cos values from 0 to 90 degrees.
  • Patterns: Notice that sin values increase from 0 to 1, while cos values decrease from 1 to 0.
  • Reciprocal Ratios: Remember that cosec is 1/sin, sec is 1/cos, and cot is 1/tan. Once you know sin, cos, and tan, you can easily find the others.
  • Tangent Relation: tan A = sin A / cos A.

Quick Revision

  • sin 30° = 1/2, cos 30° = sqrt(3)/2, tan 30° = 1/sqrt(3)
  • sin 45° = 1/sqrt(2), cos 45° = 1/sqrt(2), tan 45° = 1
  • sin 60° = sqrt(3)/2, cos 60° = 1/2, tan 60° = sqrt(3)
  • Reciprocal ratios are essential for cosec, sec, and cot.

Practice Questions

  1. Evaluate: sin 60° + cos 30°
  2. If 2 sin θ = sqrt(3), find the value of θ (theta).
  3. Find the value of (tan 30° + tan 60°) / (1 + tan 30° * tan 60°).
  4. Is sin 30° + sin 60° = sin 90°? Justify your answer.
  5. Calculate: (sec 45°)^2 – (tan 45°)^2