What are the Six Trigonometric Functions and Their Uses?
The concept of trigonometric functions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From calculating heights and distances in geometry to solving periodic problems in science and engineering, trigonometric functions are essential tools for students from Class 9, 10, 11, and onwards. This topic is especially important for board exams and competitive tests like JEE or NEET.
What Is Trigonometric Functions?
A trigonometric function is defined as a mathematical function that relates an angle of a right triangle to the ratios of its sides. You’ll find this concept applied in areas such as geometry, physics (like understanding waves and oscillations), and engineering. The six main trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each function describes a different relationship using the sides of a right-angled triangle.
Key Formula for Trigonometric Functions
Here’s the standard formulae for the six trigonometric functions for an angle θ in a right-angled triangle:
| Function | Formula |
|---|---|
| sin θ | Opposite / Hypotenuse |
| cos θ | Adjacent / Hypotenuse |
| tan θ | Opposite / Adjacent |
| cosec θ | Hypotenuse / Opposite |
| sec θ | Hypotenuse / Adjacent |
| cot θ | Adjacent / Opposite |
Some of the most commonly used trigonometric identities include:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Standard Values of Trigonometric Functions
| Degrees | Radians | sin θ | cos θ | tan θ | cosec θ | sec θ | cot θ |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | Undefined | 1 | Undefined |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | π/2 | 1 | 0 | Undefined | 1 | Undefined | 0 |
Cross-Disciplinary Usage
Trigonometric functions are not only useful in Mathematics, but also play an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see trigonometry used in projectile motion, alternating current, and digital signal processing questions.
Step-by-Step Illustration
Let’s solve an example using the tangent function:
1. A boy is standing 10 metres from a tree. The angle of elevation to a bird on top is 20°.2. Let x be the height from ground to the bird.
3. Formula: tan θ = Opposite / Adjacent
4. Substitute the values: tan(20°) = x / 10
5. x = 10 × tan(20°) ≈ 10 × 0.364 = 3.64 meters
6. Final Answer: The bird is sitting approx. 3.6 m from the ground.
Speed Trick or Vedic Shortcut
Here’s a quick way to remember the three main trigonometric ratios: SOH-CAH-TOA.
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
This trick is used by students for fast recall in competitive and board exams.
Try These Yourself
- Find sin 30°, sin 45°, sin 60° using the value table above.
- If tan θ = 1, what is the value of θ between 0° and 90°?
- Is sec 90° defined?
- Write the cofunction identity for cos(90° – θ).
Frequent Errors and Misunderstandings
- Confusing the numerator and denominator in the main trigonometric ratios.
- Not knowing that tan 90° (and similar values) can be undefined.
- Mixing degrees and radians when using calculators.
Relation to Other Concepts
The idea of trigonometric functions connects closely with topics such as the unit circle and trigonometric identities. Mastering this helps with understanding graphs, calculus (like differentiation and integration of trig functions), and many advanced chapters in school Maths and competitive exams. For further study, visit Trigonometric Ratios and Trigonometric Equations.
Classroom Tip
A quick way to remember trigonometric functions is with the SOH-CAH-TOA rhyme—and by repeatedly practicing value tables for common angles (like 0°, 30°, 45°, and 60°). Vedantu’s teachers often use colored charts and fun mnemonics to simplify learning during live classes.
We explored trigonometric functions—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept. For more, check related topics like Applications of Trigonometry, Trigonometry Overview, and Trigonometry Table.
FAQs on Trigonometric Functions in Maths: Definition, Formulas & Applications
1. What are the six fundamental trigonometric functions in Maths?
The six fundamental trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). In a right-angled triangle, these are defined as ratios of its sides (opposite, adjacent, hypotenuse) with respect to an angle θ. For a broader definition applicable to any angle, they are defined using the coordinates of a point on the unit circle.
2. What is the main difference between trigonometric ratios and trigonometric functions?
The primary difference lies in their domain and application. Trigonometric ratios are strictly defined for acute angles (0° to 90°) within a right-angled triangle. Trigonometric functions are a more general concept, extending these ratios to any real number angle (positive, negative, or greater than 360°) using the unit circle. This allows them to have properties like periodicity and be represented as continuous waves on a graph.
3. Why is the unit circle so important for understanding trigonometric functions?
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is crucial because it provides a visual and comprehensive definition for sine and cosine for all angles. For any angle θ, the x-coordinate of the point on the circle is cos(θ) and the y-coordinate is sin(θ). This helps in easily determining the sign of trigonometric functions in different quadrants and understanding their periodic nature.
4. In trigonometry, why is it often preferred to measure angles in radians instead of degrees?
While degrees are a common unit, radians are often preferred in higher mathematics for a key reason: they are a more natural and dimensionless measure of an angle. A radian is directly related to the radius and arc length of a circle (Angle in radians = Arc Length / Radius). This direct relationship simplifies many important formulas in calculus and physics, such as the derivatives and integrals of trigonometric functions, making calculations more straightforward.
5. How can we determine the signs (+ or -) of trigonometric functions in the four different quadrants?
The signs of trigonometric functions depend on the quadrant in which the angle terminates. This can be remembered using the ASTC rule:
- Quadrant I (0° to 90°): All functions are positive.
- Quadrant II (90° to 180°): Sine (and its reciprocal, cosecant) are positive.
- Quadrant III (180° to 270°): Tangent (and its reciprocal, cotangent) are positive.
- Quadrant IV (270° to 360°): Cosine (and its reciprocal, secant) are positive.
6. What are the three fundamental Pythagorean identities in trigonometry?
The Pythagorean identities are derived from the Pythagorean theorem on the unit circle and are fundamental for simplifying expressions and solving equations. The three main identities are:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
7. What are some key real-world applications of trigonometric functions?
Trigonometric functions are essential in numerous fields for modeling periodic phenomena and solving problems related to angles and distances. Key applications include:
- Physics: Modeling sound waves, light waves, oscillations of a pendulum, and alternating currents.
- Engineering: Designing structures, calculating forces on bridges, and in electrical engineering.
- Navigation: Used in GPS, aviation, and astronomy to pinpoint locations and calculate distances between celestial bodies.
- Computer Graphics: For rotating and transforming 3D models and creating realistic animations.
8. What is the difference between a principal solution and a general solution of a trigonometric equation?
When solving a trigonometric equation, we find angles that satisfy it. The solutions can be categorized as:
- Principal Solution: The solution that lies within a restricted, specific range. For sine and tangent, this is typically [-π/2, π/2], and for cosine, it is [0, π]. It provides a single, unique answer.
- General Solution: An expression that represents all possible solutions to the equation. Since trigonometric functions are periodic, there are infinite solutions. The general solution includes a term 'n' (an integer) to account for this repetition, e.g., x = α + 2nπ.
9. How do the graphs of sine and cosine functions compare?
The graphs of sine (y = sin x) and cosine (y = cos x) are very similar, both being periodic waves known as sinusoids. The main characteristics are:
- Shape: Both are smooth, continuous wave-like curves.
- Amplitude: Both have an amplitude of 1, meaning their values oscillate between -1 and 1.
- Period: Both have a period of 2π, meaning the graph repeats every 2π radians.
- Phase Shift: The key difference is a phase shift. The graph of cosine is the graph of sine shifted π/2 units to the left. That is, cos(x) = sin(x + π/2).
10. What are inverse trigonometric functions and why is their range restricted?
Inverse trigonometric functions (like arcsin, arccos, arctan) are used to find the angle when the value of the trigonometric ratio is known. For example, if sin(θ) = 0.5, then arcsin(0.5) = θ. The range of these functions must be restricted because the original trigonometric functions are periodic (many-to-one), meaning multiple angles have the same sine or cosine value. To ensure that the inverse is a true function (one-to-one), we restrict the output to a specific interval called the principal value range, which guarantees a single, unique angle as the answer.
