How to Find Sin 15 in Fraction and Root Form?
The concept of Value of Sin 15 is a foundational trigonometric result that helps in solving trigonometry questions efficiently. It is highly relevant in board exams, entrance tests, and many real-life applications requiring angle calculations.
What Is Value of Sin 15?
The value of sin 15 refers to the exact trigonometric ratio for a 15° angle, where sine (sin) represents the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. This concept is directly applied in trigonometric calculations, geometry, and physics, and appears frequently in class 10, 11, and 12 Maths exams. You’ll find this concept applied in trigonometric values, compound angle formulas, and MCQ-based competitive exams.
Key Formula for Value of Sin 15
Here’s the standard formula:
\( \sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588 \)
Why Is Value of Sin 15 Important?
The value of sin 15 is often not memorized like sin 0°, 30°, 45°, 60°, or 90°, but questions based on it are very common. Knowing how to calculate or recall sin 15 can help you answer trigonometric problems quickly and accurately in school board exams, JEE, NEET, SAT and even real-life situations where precise angle measurement is needed.
Step-by-Step Derivation Using Compound Angle Formula
Let’s prove sin 15° step by step using the compound angle (difference) formula:
- Write 15° as (45° − 30°).
- Apply the formula: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).
- Let A = 45°, B = 30°.
\( \sin 15^\circ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ \) - Substitute standard values:
\( \sin 45^\circ = \frac{1}{\sqrt{2}} \), \( \cos 30^\circ = \frac{\sqrt{3}}{2} \),
\( \cos 45^\circ = \frac{1}{\sqrt{2}} \), \( \sin 30^\circ = \frac{1}{2} \) - Calculate:
\( = \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \times \frac{1}{2} \)
\( = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} \)
\( = \frac{\sqrt{3} - 1}{2\sqrt{2}} \) - Rationalise the denominator:
Multiply the numerator and denominator by \( \sqrt{2} \):
\( = \frac{(\sqrt{3} - 1) \times \sqrt{2}}{2\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{6} - \sqrt{2}}{4} \) - Decimal approximation:
\( \sqrt{6} \approx 2.449 \), \( \sqrt{2} \approx 1.414 \), so
\( \frac{2.449 - 1.414}{4} = \frac{1.035}{4} \approx 0.2588 \)
Sin 15 in Root, Fraction and Decimal Forms
| Form | Value | Remarks |
|---|---|---|
| Root Form | \( \frac{\sqrt{6} - \sqrt{2}}{4} \) | Exact value, suits board exam proofs |
| Fraction (+rationalised root) | \( \frac{(\sqrt{3} - 1)}{2\sqrt{2}} \) | Also accepted in exams |
| Decimal | 0.2588 | Up to 4 decimal places |
Visual Representation: Sin 15 on Unit Circle
On the unit circle, the value of sin 15° is the y-coordinate of the point corresponding to a 15° angle from the positive x-axis. Since 15° is in the first quadrant, sin 15° is positive. Its approximate value (0.2588) means that for a radius of 1, the height from the x-axis up to the curve at 15° is 0.2588 units.
Trigonometric Values Table (Quick Revision)
| Angle (°) | sin | cos | tan |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 15 | \( \frac{\sqrt{6} - \sqrt{2}}{4} \) | \( \frac{\sqrt{6} + \sqrt{2}}{4} \) | 0.2679 |
| 30 | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | 0.5774 |
| 45 | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{\sqrt{2}} \) | 1 |
| 60 | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | 1.732 |
| 90 | 1 | 0 | undefined |
Speed Trick or Vedic Shortcut
You don't have to memorize sin 15 as a new value—just break it into known angles (like 45° and 30°) and use the compound angle formula. If you forget that, just remember this pattern: “sin(small angle) = sin(known – known)”. Most students know sin 45 and sin 30, so combine them as shown above to save exam time.
Common Mistakes and Tips
- Forgetting to rationalise the denominator.
- Mixing the order of subtraction in the numerator.
- Using decimal approximations instead of exact values in proofs.
- Confusing sin 15° with sin 150° (which equals 1/2).
- Not remembering which quadrant 15° belongs to (it's positive).
Relation to Other Values and Formulas
Value of cos 15 is \( \frac{\sqrt{6} + \sqrt{2}}{4} \); both are useful when solving length, height, or distance problems. Also note that:
sin 150° = sin(180° − 30°) = sin 30° = 1/2,
sin 15 is much less than sin 30,
sin 75° = cos 15°.
Mastering these values helps with many trigonometry applications and conversions between different angle measures.
Try These Yourself
- Find the value of sin 75° in exact form.
- Prove that cos 15° = (√6 + √2)/4.
- If sin A = sin 15°, what is A in degrees (for 0 ≤ A < 180°)?
- Simplify: 2 × sin 15° × cos 15°.
Wrapping It All Up
We explored the value of sin 15—from its definition to the proof, shortcut tips, mistakes, and how it connects to related formulas. With these methods, you’ll be able to answer trigonometry questions more confidently and accurately. Practice with Vedantu for more tips and detailed explanations that make exam prep easier!
Further Reading and Related Topics
FAQs on Value of Sin 15°: Derivation, Formula & Tricks
1. What is the value of sin 15°?
The value of sin 15° is (√6 - √2) / 4, which is approximately equal to 0.2588. This value is often used in trigonometry problems and can be derived using the compound angle formula.
2. How can I find sin 15° using a formula?
You can derive the value using the compound angle formula: sin(A - B) = sin A cos B - cos A sin B. By substituting A = 45° and B = 30°, you get: sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (1/√2)(√3/2) - (1/√2)(1/2) = (√3 - 1) / (2√2). Rationalizing the denominator gives you the final answer: (√6 - √2) / 4.
3. What is the value of sin 15° in decimal form?
The decimal approximation of sin 15° is approximately 0.2588.
4. How is sin 15° related to other trigonometric values?
sin 15° is related to other angles through trigonometric identities. For example, sin 15° = cos 75° (complementary angles), and it's involved in various compound angle calculations. Understanding its relationship with 30°, 45°, and their multiples is crucial.
5. What is the value of cos 15°?
The value of cos 15° is (√6 + √2) / 4, which is approximately 0.9659. This is derived similarly using the compound angle formula, but with the addition instead of subtraction.
6. Can I use a calculator to find sin 15°?
While a calculator can give you the decimal approximation quickly, understanding the derivation is essential for exams where calculators might be restricted. Knowing the exact value in root form ((√6 - √2) / 4) is also beneficial.
7. How is sin 15° used in problem-solving?
sin 15° appears in many trigonometry problems involving triangles, especially those requiring the application of the sine rule or cosine rule. It also surfaces in more advanced topics like wave equations and complex numbers.
8. Is sin 15° a rational or irrational number?
sin 15° is an irrational number because its exact value contains square roots that cannot be expressed as a simple fraction.
9. What are some common mistakes students make when calculating sin 15°?
Common errors include incorrect application of the compound angle formula, sign errors during calculations, and mistakes in simplifying the surds (roots). Double-checking your steps and understanding the formula's derivation helps minimize these errors.
10. How can I remember the value of sin 15° easily?
Try associating the value with the formula's structure and the known values of sin 30°, sin 45°, cos 30°, and cos 45°. Regular practice and visualization using the unit circle will improve your recall.
11. What is the relationship between sin 15° and sin 165°?
sin 165° and sin 15° are related through the identity sin(180° - x) = sin x. Therefore, sin 165° = sin (180° - 15°) = sin 15° = (√6 - √2) / 4
