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