What is the value of sin 60 degrees in fraction, decimal, and radians?
The concept of sin 60 degrees plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From geometry to trigonometry and physics, knowing the exact value of sin 60 degrees helps solve triangles, word problems, and even physics questions with speed and accuracy.
What Is Sin 60 Degrees?
Sin 60 degrees is a trigonometric value representing the ratio of the length of the side opposite a 60° angle to the hypotenuse in a right-angled triangle. You’ll find this value applied in topics such as trigonometric ratios, the unit circle, and geometry word problems.
Key Formula for Sin 60 Degrees
Here’s the standard formula: \( \sin 60^\circ = \frac{\sqrt{3}}{2} \) or approximately \( 0.866 \)
| Angle (Degrees) | Angle (Radians) | Sin Value (Fraction) | Sin Value (Decimal) |
|---|---|---|---|
| 60° | π/3 | √3/2 | 0.866 |
Cross-Disciplinary Usage
Sin 60 degrees is not only useful in Maths but also plays an important role in Physics, Computer Science, and logical reasoning. Students preparing for JEE, NEET, and board exams often encounter problems that use this value for solving triangles, vector decomposition, and even calculation of heights and distances. It’s also a basic building block for advanced trigonometry, as well as other standard trigonometric angles.
Step-by-Step Illustration
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Draw an equilateral triangle with all sides = 2 units. Drop a height from one vertex to the base.
This height splits the base into 1 unit + 1 unit and creates two 30-60-90 right triangles.
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Calculate the height using Pythagoras’ Theorem:
Let height = h.
\( h^2 + 1^2 = 2^2 \Rightarrow h^2 = 4 - 1 = 3 \Rightarrow h = \sqrt{3} \) -
In the right-angled triangle, for angle 60°:
Opposite = h = √3, Hypotenuse = 2
\( \sin 60^\circ = \dfrac{\text{Opposite}}{\text{Hypotenuse}} = \dfrac{\sqrt{3}}{2} \) - Decimal value: \( \sin 60^\circ \approx 0.866 \)
Sin 60 Degrees in Trigonometric Table
| Angle | Sin Value | Cos Value | Tan Value |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Sin 60 Degrees in Different Forms
| Form | Sin 60° Value |
|---|---|
| Fraction/Surd | √3/2 |
| Decimal | 0.866 |
| Radian form | sin(π/3) |
| Unit circle (coordinates) | (½, 0.866) |
Applications of Sin 60 Degrees
You’ll use sin 60 degrees in lots of practical problems. For example, calculating the height of a triangle given the hypotenuse, working with trigonometric ratios, or decomposing forces at a 60° angle in physics. It’s also vital for MCQ accuracy and last-minute revision before competitive exams like JEE and CBSE boards.
Solved Example:
Find the perpendicular height of an equilateral triangle of side 4 cm.
1. Each side = 4 cm. Height splits base into 2 cm.
2. By Pythagoras’ Theorem,
\( h^2 + 2^2 = 4^2 \implies h^2 = 16 - 4 = 12 \implies h = 2\sqrt{3} \) cm
So, height = \( 2\sqrt{3} \) cm = 3.464 cm (using sin 60° value)
Speed Trick or Vedic Shortcut
Here’s a fast recall trick: The value of sin 60° is always larger than sin 30° and sin 45° but less than sin 90°. A memory hack is “increasing order for sin: 30° (½), 45° (1/√2 ≈ 0.707), 60° (√3/2 ≈ 0.866), 90° (1).” Vedic methods and quick tables are shared in Vedantu’s live coaching for rapid revision and MCQ speed.
Try These Yourself
- Derive sin 60° using a triangle with sides 2 units and a height.
- Find the value of sin 60° + cos 30°.
- If sin A = √3/2, what is the value of angle A?
- Express sin 60 degrees in terms of tan 60 degrees.
Frequent Errors and Misunderstandings
- Mixing up sin 60 degrees (√3/2) with sin 30 degrees (½).
- Confusing surd and decimal values during MCQ exams.
- Incorrect use of calculators: forgetting to switch to degree/radian mode.
- Using the trigonometric table for the wrong quadrant or angle.
Relation to Other Concepts
The idea of sin 60 degrees connects closely to sin 30 degrees, sin 90 degrees, and cos 60 degrees. In fact, sin 60° = cos 30°, and these relations help when applying complementary angle formulas or working with the trigonometric table. Mastering sin 60 also makes right triangle and unit circle concepts clear for future topics.
Classroom Tip
A quick way to remember sin 60 degrees is to think of an equilateral triangle and realize: dropping its height always forms a 30-60-90 triangle, and the exact value pops right out as √3/2. Vedantu’s teachers often share such visual and mnemonic tricks so you can learn and recall faster during online live classes.
We explored sin 60 degrees—from its definition, key formula, derivation, applications, and common mistakes. With practice and the right memory hacks, this value will become second nature for you in exams. Keep reviewing with Vedantu to boost your confidence in trigonometry and related maths topics!
Related Links for Quick Revision:
FAQs on Sin 60 Degrees – Exact Value, Formula & Uses with Examples
1. What is the value of sin 60 degrees?
The exact value of sin 60 degrees is √3/2, which is approximately equal to 0.866. This value is crucial in various trigonometric calculations and applications.
2. How do you derive the value of sin 60 degrees?
Consider an equilateral triangle with sides of length 2 units. Dropping an altitude from one vertex bisects the base and creates two 30-60-90 right-angled triangles. The altitude has length √3 units (by the Pythagorean theorem). Therefore, sin 60° (opposite/hypotenuse) = √3/2.
3. What is sin 60 degrees in decimal form?
Sin 60 degrees in decimal form is approximately 0.8660254. This decimal approximation is often used in calculations where a precise surd form is not required.
4. How is sin 60 degrees represented on the unit circle?
On the unit circle, a 60° angle from the positive x-axis intersects the circle at the point (1/2, √3/2). The y-coordinate, √3/2, represents the value of sin 60°.
5. What are some common applications of sin 60 degrees?
Sin 60° is frequently used in solving problems related to:
- Geometry: Calculating lengths of sides and heights in triangles.
- Physics: Resolving vectors, calculating forces, and analyzing motion.
- Engineering: Structural calculations, design, and analysis.
6. Is sin 60 degrees equal to sin 120 degrees?
While both sin 60° and sin 120° have the same numerical value (√3/2), they represent different angles with different geometric interpretations. Sin 120° is equivalent to sin (180° - 60°).
7. What is the relationship between sin 60 degrees and cos 30 degrees?
Sin 60° and cos 30° are equal (√3/2). This is a consequence of the complementary angle identity: sin θ = cos (90° - θ).
8. How can I quickly memorize the value of sin 60 degrees?
Use mnemonics or visualization techniques. Remembering the 30-60-90 triangle derivation can help. Practicing problems frequently reinforces memory.
9. Are there any common mistakes students make when working with sin 60 degrees?
Common mistakes include confusing sin 60° with sin 30° (1/2) or incorrectly using the decimal approximation. Careful calculation and understanding the concept is key.
10. How does sin 60 degrees relate to radians?
60 degrees is equivalent to π/3 radians. Therefore, sin (π/3) = √3/2.
11. Can sin 60 degrees be expressed in surd form?
Yes, the surd form of sin 60 degrees is √3/2. This is the exact value and often preferred in mathematical calculations to avoid rounding errors.
12. Where can I find more practice problems involving sin 60 degrees?
Vedantu provides numerous practice problems, worksheets, and solved examples related to trigonometric functions like sin 60°. Check our website and app for more resources.
