How to Calculate Sin 1 Degree (Step-by-Step)
The concept of Sin 1 is a key element in trigonometry, often used in board exams, competitive test preparation, and real-life measurement scenarios. Understanding Sin 1 helps students handle questions involving small angles and quick trigonometric substitutions for faster problem-solving.
What Is Sin 1?
Sin 1 refers to the value of the sine function when its input (angle) is either 1 degree or 1 radian. In trigonometry, the sine of an angle represents the ratio of the length of the side opposite the angle to the hypotenuse in a right-angled triangle. Sin 1 is used for both degree and radian modes, so it's important to specify which unit is being used. You’ll find this concept applied in trigonometric calculations, coordinate geometry problems, and even in certain physics applications such as wave functions.
Key Formula for Sin 1
Here’s the standard formula for the sine of an angle:
\( \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} \)
For small angles:
- Sin 1° (sine of 1 degree) ≈ 0.01745
- Sin 1 (in radians) ≈ 0.84147
Value Table for Sin 1 (Degree vs Radian)
| Angle | Unit | Sin Value (Decimal) |
|---|---|---|
| 1 Degree | Degrees (°) | 0.01745 |
| 1 Radian | Radians | 0.84147 |
How to Calculate Sin 1 Step-by-Step
- Decide if you want Sin 1 degree or Sin 1 radian.
- If using a calculator, set the correct mode (DEG for degrees, RAD for radians).
- For Sin 1°, input 1 and press the “sin” button → You get 0.01745.
- For Sin 1 (radian), input 1 (RAD mode) → You get 0.84147.
- Manual method (Taylor expansion for small angles, radians):
\( \sin x ≈ x - \frac{x^3}{6} + \frac{x^5}{120} \)
For x = 1 (radian):
\( \sin 1 ≈ 1 - \frac{1^3}{6} + \frac{1^5}{120} \) ≈ 1 - 0.16667 + 0.00833 ≈ 0.84167
Sin 1 vs Sin-1(1) (Inverse Sine)
| Expression | Meaning | Value |
|---|---|---|
| sin 1 | Sine of 1 (angle in degree or radian) | 0.01745 (°), 0.84147 (rad) |
| sin-1(1) | Angle whose sine is 1 (inverse sine) | 90° (π/2) |
Sin-1(1), also known as arcsin(1), gives the angle whose sine is 1. This is always 90° or π/2 radians. Don’t confuse this with “1 over sin”, which means cosec.
Cross-Disciplinary Usage
Sin 1 isn’t just useful in Mathematics. You often use it in Physics (oscillations, waves), Engineering (signal processing), and even Computer Science for angle and graphics calculations. Students targeting competitions like JEE will use Sin 1 and inverse trigonometric values in problem-solving and MCQs.
Applications of Sin 1
- Solving right triangle and geometry problems where non-standard angles arise.
- Estimating small displacements and angle-based calculations in Physics lab work.
- Quick substitutions in competitive exam questions to avoid calculator dependency.
- Used for understanding approximation methods in Calculus (Taylor series, small angle approximations).
Step-by-Step Illustration: Example Problem
Example: Find 2 × sin 1 radian, step by step.
1. Write the expression: 2 × sin 12. Calculate sin 1 (in radians): sin 1 ≈ 0.84147
3. Multiply by 2: 2 × 0.84147 = 1.68294
Final Answer: 2 × sin 1 ≈ 1.68
Speed Tip: Quick Use of Sin 1
For small angles in degrees (less than about 10°), you can use the approximation:
sin θ ≈ θ (when θ is in radians)
So, for 1°, first convert to radians:
1° = π/180 ≈ 0.01745 radians
Therefore, sin 1° ≈ 0.01745 (which matches calculator value!). This trick helps you check answers rapidly during exams.
Try These Yourself
- Find sin 1° using the Taylor series up to 3 terms.
- Is sin 1 more accurate in degree or radian for small angle approximation?
- Calculate sin-1(1). What does your calculator say?
- Find 4 × sin-1(1), and write the answer in radians.
Frequent Errors and Misunderstandings
- Mixing up sin 1° with sin 1 radian — always check angle units!
- Using sin-1(x) for 1/x instead of the correct inverse function.
- Not setting the calculator mode (degree/radian) correctly, causing wrong answers.
- Assuming sin 1 is a standard value like sin 30 or sin 45, but it must be approximated or looked up.
Relation to Other Concepts
Understanding Sin 1 is closely related to the study of Trigonometric Values, the properties of standard angles like Sin 30 Degrees, and other trigonometric ratios featured in Trigonometric Functions. Mastery of Sin 1 also helps with graph interpretation and Taylor series approximations for higher-level math.
Classroom Tip
To remember “sin 1 degree ≈ 0.017”, think of it as almost equal to “1/57”. For quick answers, relate 1 degree to π/180 radians. Vedantu’s tutors often use visual charts and tables to make these conversions second nature during live sessions.
We explored Sin 1—from its definition and calculation to real applications, common errors, and related trigonometry topics. Practice using these small angle values with Vyedantu’s resources for a fast boost in your problem-solving skills!
Explore further: Sin 30 Degrees, Trigonometric Values, Sine, Cosine, and Tangent, Trigonometry.
FAQs on Sin 1 – Value, Formula, Calculation & Examples
1. What is sin 1 in Maths?
In mathematics, sin 1 typically refers to the sine of an angle measured in either degrees or radians. The value differs significantly depending on the unit used. It's crucial to specify whether you're working with degrees or radians to avoid ambiguity in calculations.
2. What is the value of sin 1 degree?
The value of sin 1° is approximately 0.01745 radians. This value is often used in trigonometric calculations where angles are expressed in degrees.
3. What is the value of sin 1 radian?
The value of sin 1 radian is approximately 0.84147. Note the significant difference compared to sin 1° due to the different angular units.
4. How do I calculate sin 1 without a calculator?
Calculating sin 1 manually requires using techniques like the Taylor series expansion for the sine function. This involves an infinite series, and you'll need to truncate the series after a certain number of terms to get an approximation. The more terms you use, the greater the accuracy, but the more complex the calculation becomes. Alternatively, you could use trigonometric tables to find an approximate value.
5. What is the difference between sin 1° and sin 1 radian?
The key difference lies in the unit of measurement for the angle. 1° is a degree, while 1 radian is approximately 57.3°. Therefore, sin 1° and sin 1 radian result in vastly different numerical values. Always clearly indicate the unit (degrees or radians) when working with trigonometric functions.
6. What is sin-1(1)?
sin-1(1), also written as arcsin(1), represents the inverse sine function. It asks: 'What angle has a sine value of 1?' The answer is 90° or π/2 radians.
7. What is the difference between sin 1 and sin-1(1)?
sin 1 is the sine of an angle (either 1° or 1 radian), resulting in a numerical value. sin-1(1), or arcsin(1), is the inverse sine function, which finds the angle whose sine is 1. They are inverse operations.
8. How is sin 1 used in practical applications?
Sin 1 (in degrees or radians) finds applications in various fields, including:
- Physics: Calculating projectile motion, wave propagation, and oscillations.
- Engineering: Solving problems related to mechanics, structural analysis, and signal processing.
- Computer graphics: Used in transformations, rotations, and calculations involving 3D objects.
9. Can sin 1 be negative?
The sine of an angle is negative in the third and fourth quadrants of the unit circle. Since 1° and 1 radian are both in the first quadrant, sin 1° and sin 1 radian are both positive.
10. What is the Taylor series expansion for sin x, and how can I use it to approximate sin 1?
The Taylor series expansion for sin x (where x is in radians) is: sin x ≈ x - x³/3! + x⁵/5! - x⁷/7! + ... To approximate sin 1, substitute x = 1 radian into the series. The more terms you include, the more accurate the approximation. Remember that the series uses radians, not degrees.
11. How accurate should my approximation of sin 1 be for exams?
The required accuracy for approximating sin 1 in exams depends on the context and the marking scheme. Generally, using a calculator provides sufficient precision. If a manual calculation is required, the number of terms in the Taylor series expansion should be chosen to achieve the desired level of accuracy.
12. Where can I find a table of trigonometric values?
Trigonometric tables are available in many mathematics textbooks and online resources. These tables provide the values of trigonometric functions for various angles, including approximations for values like sin 1°. Using such tables is a faster way to find approximate values without manual calculation or using calculators.
