How to Derive and Remember Cos 30° as a Fraction?
The concept of cos 30 degrees in fraction plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Cos 30 Degrees in Fraction?
Cos 30 degrees in fraction is the exact trigonometric value of the cosine function at 30°, which is √3/2. In mathematics, cosine represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. The value is important for solving triangles, engineering, and competitive exams. Related concepts include sin 30, trigonometric values table, and standard angle proofs.
Key Formula for Cos 30 Degrees in Fraction
Here’s the standard formula:
\[
\cos(30^\circ) = \frac{\sqrt{3}}{2}
\]
Value Table: Cos 30 Degrees
| Expression | Degree | Radian | Fraction | Decimal | Radical |
|---|---|---|---|---|---|
| cos 30° | 30° | π/6 | √3/2 | 0.866 | √3/2 |
How to Derive Cos 30 Degrees in Fraction
Let’s see how to prove cos 30 degrees is √3/2 using a 30-60-90 triangle:
- Draw a right-angled triangle with one angle 30°.
- The hypotenuse is 2 units, the side opposite 30° is 1 unit, and the side adjacent to 30° is √3 units.
- Cos 30° = (adjacent side)/(hypotenuse) = √3/2.
Alternatively, using the unit circle, the x-coordinate of the point at 30° (π/6 radians) is √3/2, matching our result.
Cos 30 Degrees in Trigonometric Table
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 60° | √3/2 | 1/2 | √3 |
Applications and Exam Examples Using Cos 30 Degrees in Fraction
- Board Exam MCQs: What is cos 30° in fraction? Answer: √3/2
- Triangle Calculation: If the adjacent side is 6, find hypotenuse for 30°. 6 / hypotenuse = √3/2 ⇒ hypotenuse = 6 × 2 / √3 = 4√3
- Unit Circle Problems: Find the x-coordinate at 30°. x = √3/2
Speed Trick or Classroom Tip
Remember: For 30°, cos is always √3/2 and sin is 1/2. For 60°, these swap. Vedantu’s teachers often suggest the mnemonic “Short, Tall—Sin, Cos—all” to keep angle values clear for exams.
Frequent Errors and Misunderstandings
- Confusing cos 30° (√3/2) with sin 30° (1/2).
- Forgetting to write the denominator 2 under the square root 3.
- Using decimal instead of fraction in exact calculations.
- Mixing up radians and degrees.
Relation to Other Concepts
The idea of cos 30 degrees in fraction connects to cos 60 degrees, trigonometric ratio tables, and the unit circle. Understanding this helps you solve geometry, algebra, and physics problems efficiently.
Quick FAQ on Cos 30 Degrees in Fraction
- What is the value of cos 30 degrees in fraction? √3/2
- Is cos 30 equal to 1/2? No, that’s sin 30°. Cos 30° is √3/2.
- Decimal value of cos 30°? About 0.866.
- Is cos 30° used in exams? Yes, very commonly for triangles and trigonometric calculations.
- How to remember cos 30°? Adjacent over hypotenuse in a 30-60-90 triangle is always √3/2.
We explored cos 30 degrees in fraction—from definition, formula, derivation, trigonometry tables, mistakes, and connections to other subjects. Continue practicing with Vedantu to strengthen your skills in trigonometry and maths overall!
Related Reading: Cos 60 Degrees Value | Sin 30 Degrees | Standard Angles Table | Trigonometry Full Guide
FAQs on What is the Value of Cos 30 Degrees in Fraction?
1. What is the exact value of cos 30 degrees in fraction form?
The exact value of cos 30 degrees in fraction form is √3/2. This is an irrational number, which is approximately 0.866 in decimal form. In a right-angled triangle, it represents the ratio of the length of the adjacent side to the hypotenuse for a 30° angle.
2. How is the value of cos 30° derived as per the NCERT syllabus?
The value of cos 30° (√3/2) can be derived using two main methods taught in the CBSE/NCERT curriculum:
- Using a 30-60-90 Triangle: Start with an equilateral triangle of side length 2. Bisecting one angle creates two 30-60-90 right-angled triangles. The sides of each new triangle are 1, √3, and 2. For the 30° angle, the adjacent side is √3 and the hypotenuse is 2, so cos 30° = adjacent/hypotenuse = √3/2.
- Using the Unit Circle: On a unit circle (radius of 1), a point on the circumference at an angle of 30° has coordinates (x, y). The x-coordinate represents the cosine value. For 30°, this x-coordinate is √3/2.
3. Is the value of cos 30° the same as sin 30°?
No, they have different values. This is a common point of confusion. The value of cos 30° is √3/2 (approx. 0.866), whereas the value of sin 30° is 1/2 (0.5). In a right-angled triangle, cos 30° relates the adjacent side to the hypotenuse, while sin 30° relates the opposite side to the hypotenuse.
4. Why is the value of cos 30° positive?
The value of cos 30° is positive because the angle 30° lies in the first quadrant (from 0° to 90°) of the Cartesian coordinate system. According to the CAST rule in trigonometry, all trigonometric functions (Sine, Cosine, Tangent) are positive in the first quadrant.
5. How does the value of cos 30° compare to cos 60°?
The values are different but related through the concept of complementary angles. The value of cos 30° is √3/2, while cos 60° is 1/2. The relationship is that cos(θ) = sin(90° - θ). Therefore, cos 30° = sin(90° - 30°) = sin 60°, which is also √3/2.
6. Where is the value of cos 30° used in a practical, real-world example?
In physics and engineering, cos 30° is frequently used to resolve forces. For instance, if you are pushing a lawnmower with the handle at a 30° angle to the ground, the effective horizontal force pushing the mower forward is calculated using cos 30°. It helps determine how much of your effort contributes to the forward motion.
7. What is an easy way to remember the value of cos 30° for exams?
A simple trick is to remember the standard trigonometric values for sine and cosine. For angles 0°, 30°, 45°, 60°, and 90°, the numerators for cosine follow the pattern √4, √3, √2, √1, √0, all divided by 2. For 30°, this gives you √3/2. Visualising the 30-60-90 triangle also helps, where the side adjacent to 30° is √3 and the hypotenuse is 2.
8. What is the equivalent of 30 degrees in radians?
30 degrees is equivalent to π/6 radians. To convert any angle from degrees to radians, you multiply the degree measure by π/180. For 30°, the calculation is 30 × (π/180), which simplifies to π/6.
