Class 10 Notes - Introduction to Trigonometry

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Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles, especially right-angled triangles. It is widely used in geometry, physics, engineering, astronomy, and many other fields.

1. What is Trigonometry?

  • The word "Trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure).
  • It deals with the measurement of angles and the calculation of lengths in triangles.
  • Trigonometry is mainly concerned with right-angled triangles.

2. Trigonometric Ratios

In a right-angled triangle, the sides are named as:

  • Hypotenuse: The side opposite the right angle (the longest side).
  • Opposite side: The side opposite to the angle being considered.
  • Adjacent side: The side next to the angle being considered (other than the hypotenuse).

The six trigonometric ratios are:

  • Sine (sin): sin θ = Opposite / Hypotenuse
  • Cosine (cos): cos θ = Adjacent / Hypotenuse
  • Tangent (tan): tan θ = Opposite / Adjacent
  • Cosecant (csc or cosec): cosec θ = Hypotenuse / Opposite
  • Secant (sec): sec θ = Hypotenuse / Adjacent
  • Cotangent (cot): cot θ = Adjacent / Opposite

3. Values of Trigonometric Ratios for Standard Angles

Angle (θ) 0° 30° 45° 60° 90°
sin θ 0 1/2 1/√2 √3/2 1
cos θ 1 √3/2 1/√2 1/2 0
tan θ 0 1/√3 1 √3 Not defined

4. Some Important Trigonometric Identities

  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ

5. Applications of Trigonometry

  • Finding heights and distances
  • Navigation and astronomy
  • Engineering and construction
  • Physics problems involving angles

6. Example Problem

Problem: In a right-angled triangle, if one of the angles is 30° and the hypotenuse is 10 cm, find the length of the side opposite to the 30° angle.

Solution: sin 30° = Opposite / Hypotenuse ⇒ 1/2 = Opposite / 10 ⇒ Opposite = 10 × 1/2 = 5 cm

7. Practice Questions

  1. Define sine, cosine, and tangent for a right-angled triangle.
  2. Find the value of cos 60°.
  3. If tan θ = 1, what is the value of θ?
  4. In a right triangle, if the adjacent side is 6 cm and the hypotenuse is 10 cm, find sin θ.
  5. State the identity relating sin²θ and cos²θ.

Summary

  • Trigonometry deals with the relationship between angles and sides of triangles.
  • There are six trigonometric ratios: sin, cos, tan, cosec, sec, cot.
  • Trigonometric identities help in solving various problems.
  • Trigonometry has many real-life applications.