Introduction to Trigonometry
Understanding angles, measurements, degrees, radians and the language of trigonometry.
1. What is Trigonometry?
The word Trigonometry comes from two Greek words: “tri” meaning three, “gonia” meaning angle, and “metron” meaning measure.
In simple words, trigonometry is the branch of mathematics that studies the relationship between the angles and sides of triangles. It provides us with tools to calculate unknown sides, unknown angles, heights, distances and many quantities that are difficult to measure directly.
If you know some information about a triangle, trigonometry helps you find the information that is missing.
For example, suppose a ladder is leaning against a wall. You know the length of the ladder and the angle it makes with the ground, but you want to know how high the ladder reaches on the wall.
This is exactly the kind of problem where trigonometry becomes useful.
Trigonometry establishes relationships between the sides and angles of a triangle.
Once these relationships are understood, we can solve a huge variety of mathematical and real-world problems.
Where is Trigonometry Used?
Trigonometry is not limited to textbook triangles. It appears in many areas of mathematics, science and technology.
| Field | Application |
|---|---|
| Physics | Forces, waves, motion and oscillations |
| Engineering | Structures, mechanics and electrical systems |
| Architecture | Angles, heights and structural design |
| Astronomy | Distances and positions of celestial objects |
| Navigation | Direction, distance and position calculations |
| Computer Graphics | Rotation, movement and 3D modelling |
2. Angles and Their Measurement
Before studying trigonometric ratios, we need to understand angles.
An angle is formed when one ray rotates about a fixed point. The fixed point is called the vertex.
An angle represents the amount of rotation from one ray to another.
Consider a ray initially pointing in one direction. If it rotates anticlockwise, the angle generated is generally taken as positive. A clockwise rotation can be represented by a negative angle.
Positive and Negative Angles
In trigonometry, the direction of rotation is important.
| Direction | Convention |
|---|---|
| Anticlockwise | Positive angle |
| Clockwise | Negative angle |
A rotation of \(60^\circ\) anticlockwise represents a positive angle:
A rotation of \(60^\circ\) clockwise can be represented as:
Measuring an Angle
There are two important systems used for measuring angles:
| System | Basic Unit | Common Symbol |
|---|---|---|
| Degree System | Degree | \(^\circ\) |
| Radian System | Radian | \(\text{rad}\) |
3. Degrees and Radians
Degree Measure
In the degree system, one complete revolution is divided into 360 equal parts.
Therefore, a straight angle is half of a complete revolution:
Similarly, a right angle is one-fourth of a complete revolution:
| Angle | Fraction of Revolution |
|---|---|
| \(90^\circ\) | \(\frac{1}{4}\) |
| \(180^\circ\) | \(\frac{1}{2}\) |
| \(270^\circ\) | \(\frac{3}{4}\) |
| \(360^\circ\) | 1 complete revolution |
Radian Measure
The radian is the standard unit of angular measurement used extensively in higher mathematics, calculus and physics.
To understand a radian, imagine a circle with centre \(O\) and radius \(r\). An angle of one radian is formed when the arc intercepted by the angle has a length equal to the radius of the circle.
One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of that circle.
If the arc length is \(s\) and the radius is \(r\), then the radian measure of the angle is:
where \(\theta\) is measured in radians.
The radian measure does not depend on the size of the circle. It depends only on the ratio of arc length to radius.
Why Does \(180^\circ = \pi\) Radians?
This is one of the most important facts in Class 11 trigonometry.
For a circle of radius \(r\), the circumference is:
One complete revolution therefore corresponds to an arc length of \(2\pi r\).
Using the radian formula:
For one complete revolution:
Therefore:
But one complete revolution is also \(360^\circ\). Hence:
Dividing both sides by \(2\):
4. Degree–Radian Conversion
Since \(180^\circ = \pi\) radians, we can easily convert an angle from degrees to radians or from radians to degrees.
Degrees → Radians
To convert an angle from degrees into radians, multiply it by:
Therefore:
Multiply by \(\frac{\pi}{180}\):
Simplifying:
Therefore:
Hence:
Radians → Degrees
To convert radians into degrees, multiply by:
Therefore:
Cancel \(\pi\):
Therefore:
Important Conversion Table
| Degrees | Radians |
|---|---|
| \(0^\circ\) | \(0\) |
| \(30^\circ\) | \(\frac{\pi}{6}\) |
| \(45^\circ\) | \(\frac{\pi}{4}\) |
| \(60^\circ\) | \(\frac{\pi}{3}\) |
| \(90^\circ\) | \(\frac{\pi}{2}\) |
| \(120^\circ\) | \(\frac{2\pi}{3}\) |
| \(135^\circ\) | \(\frac{3\pi}{4}\) |
| \(150^\circ\) | \(\frac{5\pi}{6}\) |
| \(180^\circ\) | \(\pi\) |
| \(270^\circ\) | \(\frac{3\pi}{2}\) |
| \(360^\circ\) | \(2\pi\) |
For common angles, remember:
Once these are memorised, many trigonometry questions become much faster.
5. Radian Measure and Arc Length
Radian measure becomes particularly useful when we work with the length of an arc of a circle.
If an angle \(\theta\) is measured in radians, the corresponding arc length \(s\) is:
where:
- \(s\) = arc length
- \(r\) = radius of the circle
- \(\theta\) = angle in radians
A circle has radius \(7\) cm and the angle subtended at the centre is \(\frac{\pi}{3}\) radians. Find the arc length.
Using:
Substitute the values:
Therefore:
The formula \[ s=r\theta \] requires \(\theta\) to be measured in radians.
Do not directly substitute a degree value into this formula.
🚀 Section 1 — Quick Revision
- Trigonometry studies relationships between the sides and angles of triangles.
- Anticlockwise rotation is generally taken as positive.
- Clockwise rotation is generally taken as negative.
- One complete revolution is \[ 360^\circ=2\pi\text{ rad}. \]
- Therefore: \[ 180^\circ=\pi\text{ rad}. \]
- Degrees to radians: \[ \theta^\circ\times\frac{\pi}{180}. \]
- Radians to degrees: \[ \theta\times\frac{180^\circ}{\pi}. \]
- Arc length: \[ s=r\theta. \]
- In \(s=r\theta\), the angle \(\theta\) must be in radians.
