What is the Sine Function in Maths?
The concept of Sine Function plays a key role in mathematics and is widely applicable to both real-life situations (like waves and sound) and exam scenarios such as trigonometry in school or entrance tests. Understanding sine not only helps you solve triangles but also makes it easy to draw and analyze graphs in algebra and physics.
What Is Sine Function?
A sine function is defined as the trigonometric function that relates the angle of a right triangle to the ratio of the side opposite that angle over the hypotenuse. You’ll find this concept applied in areas such as geometry, sound waves, physics, and engineering. In short, for any angle θ in a right triangle, sin(θ) = (opposite side) / (hypotenuse).
Key Formula for Sine Function
Here’s the standard formula: \( \sin \theta = \dfrac{\text{Opposite Side}}{\text{Hypotenuse}} \)
Sine Function Properties & Table
The sine function is a periodic, smooth, and continuous function. Here are its key properties:
- Sine is periodic: its value repeats every 360° (2π radians).
- Its maximum value is 1, and minimum is -1 (so the range is [-1, 1]).
- Sine is an odd function: sin(-x) = -sin(x).
- The domain is all real numbers (−∞, ∞).
Here is a quick table of common angles:
| Angle (Degrees) | Angle (Radians) | sin(θ) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 1/2 |
| 45° | π/4 | 1/√2 |
| 60° | π/3 | √3/2 |
| 90° | π/2 | 1 |
| 180° | π | 0 |
| 270° | 3π/2 | -1 |
| 360° | 2π | 0 |
Graph of Sine Function
The sine function graph is a smooth, wavy curve known as the sine wave. It starts from zero, rises to 1 at 90°, falls back to zero at 180°, drops to -1 at 270°, and returns to zero at 360°. The pattern keeps repeating (period = 360° or 2π radians). You can use the graph to visualize sound waves, tides, and alternating current in physics.
Cross-Disciplinary Usage
The sine function is not only useful in Maths but also plays an important role in Physics (like understanding waves), Computer Science (signal processing), and fields like engineering and astronomy. Students preparing for JEE or NEET will see its relevance in various questions, especially those involving triangles and oscillations.
Step-by-Step Illustration: Solving a Sine Problem
Let’s solve for the length of a side using sine:
1. Suppose a right triangle has an angle θ = 30° and hypotenuse = 10 units.2. Use the formula: sin θ = (opposite side) / (hypotenuse).
3. So, sin 30° = x / 10.
4. Value of sin 30° = 1/2.
5. So, 1/2 = x / 10 ⟹ x = 10 × 1/2 = 5 units.
Final Answer: The side opposite 30° is 5 units.
Speed Trick or Vedic Shortcut
Here’s a quick trick: Remember the SOH-CAH-TOA mnemonic for all trigonometric ratios. For the sine function, “SOH” stands for “Sine = Opposite over Hypotenuse.” This helps you recall the formula instantly during exams.
Example Trick: In triangle-based MCQs, if you see the angle and the hypotenuse, just multiply sin(angle) by hypotenuse to get the opposite side—no need to redraw the triangle each time.
Try These Yourself
- Find the value of sin(45°).
- If the opposite side is 8 units and hypotenuse is 10 units, what’s sin(θ)?
- What is the amplitude and period of y = 2sin(x)?
- Sketch one cycle of the sine graph from 0° to 360°.
Frequent Errors and Misunderstandings
- Mixing up opposite and adjacent sides in the triangle.
- Confusing the sine and cosine ratios.
- Forgetting to use degrees/radians correctly in calculator mode.
- Missing negative signs for certain quadrant values.
Relation to Other Concepts
The idea of sine function connects closely with topics such as Cosine Function and Trigonometric Functions. Mastering sine makes understanding identities, transformations, and calculus-based questions easier in higher studies.
Classroom Tip
A quick way to remember the sine function is to use the unit circle: the y-coordinate of a point on the unit circle for a given angle θ (measured from the x-axis) gives you sin(θ). Vedantu’s teachers often use this to simplify learning during live classes.
We explored sine function—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using sine and other trigonometric concepts.
Further Reading and Practice
FAQs on Sine Function: Definition, Properties, Graph & Examples
1. What is the sine function formula?
The sine function, denoted as sin θ (or sin x), is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. The formula is: sin θ = Opposite / Hypotenuse. This applies when θ is an acute angle. For angles outside 0° to 90°, the definition extends using the unit circle.
2. How do you graph the sine function?
The graph of y = sin x is a continuous wave-like curve oscillating between -1 and 1. To graph it:
• Plot key points: (0, 0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0).
• The graph repeats every 2π radians (or 360°), showing its periodicity.
• The maximum value is 1 (the amplitude), and the minimum is -1.
• The graph is symmetric about the origin, indicating it's an odd function.
3. What is the domain and range of the sine function?
The domain of the sine function is all real numbers, meaning you can input any angle (in radians or degrees). The range is [-1, 1], meaning the output of the sine function will always be between -1 and 1, inclusive.
4. What is the period of the sine function?
The period of the sine function, sin x, is 2π radians (or 360°). This means the graph repeats its pattern every 2π units.
5. What are some common applications of the sine function?
The sine function has many real-world applications, including:
• Modeling periodic phenomena like sound waves, light waves, and alternating current.
• Calculating distances and heights in trigonometry problems.
• Solving problems in physics, especially those involving oscillations and waves.
• Used in engineering for various calculations related to wave analysis.
6. How does the sine function relate to the unit circle?
The sine of an angle θ in the unit circle is defined as the y-coordinate of the point where the terminal side of the angle intersects the circle. This provides a geometrical interpretation of sine for all angles, not just acute angles in right-angled triangles.
7. What are some important sine identities?
Key sine identities include:
• sin²x + cos²x = 1 (Pythagorean identity)
• sin(2x) = 2sin(x)cos(x) (Double angle formula)
• sin(x + y) = sin x cos y + cos x sin y (Sum-to-product formula)
8. What is the inverse sine function?
The inverse sine function, denoted as sin⁻¹x or arcsin x, finds the angle whose sine is x. Its domain is [-1, 1] and its range is [-π/2, π/2].
9. How do you solve sine equations?
Solving sine equations often involves using inverse sine (arcsin), considering the periodicity of sine, and identifying all solutions within a given interval. Techniques might include algebraic manipulation, trigonometric identities, and the unit circle.
10. What are the values of sin 0°, sin 30°, sin 45°, sin 60°, and sin 90°?
These are frequently used values:
• sin 0° = 0
• sin 30° = 1/2
• sin 45° = √2/2
• sin 60° = √3/2
• sin 90° = 1
11. What is the derivative of sin x?
The derivative of sin x with respect to x is cos x.
12. What is the integral of sin x?
The indefinite integral of sin x with respect to x is -cos x + C, where C is the constant of integration.
