How to Calculate Inverse Sine (sin⁻¹ x): Steps, Formula & Value Table
The concept of Inverse Sine (sin⁻¹ x or arcsin x) plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It is especially important in trigonometry, science, and engineering wherever you need to work backwards from a sine value to an angle.
What Is Inverse Sine?
The inverse sine is defined as the function that, given a number between -1 and 1, finds the angle whose sine is that number. In other words, if sin θ = x, then θ = sin⁻¹ x or θ = arcsin x. You’ll find this concept applied in areas such as solving trigonometric equations, right triangle problems, and wave analysis.
Key Formula for Inverse Sine
Here’s the standard formula: sin-1 x = y ⇔ sin y = x, where -1 ≤ x ≤ 1, y ∈ [–π/2, π/2]
Cross-Disciplinary Usage
Inverse sine is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. It is used to determine angles in waves, oscillations, building structures, and navigation. Students preparing for JEE or NEET frequently see its use in questions involving vectors, forces, and periodic motions.
Step-by-Step Illustration: Calculating sin⁻¹(x)
To find the angle whose sine is a given value, follow these simple steps:
- Check if x is between -1 and 1.
Example: x = 1/2, which is allowed.
- Use the inverse sine formula or calculator.
sin⁻¹(1/2) = 30° or π/6 radians
- Verify the answer is within the principal range –90° to 90° (or –π/2 to π/2).
| x (sine value) | sin-1(x) (degrees) | sin-1(x) (radians) |
|---|---|---|
| -1 | -90° | –π/2 |
| -1/2 | -30° | –π/6 |
| 0 | 0° | 0 |
| 1/2 | 30° | π/6 |
| 1 | 90° | π/2 |
Speed Trick or Vedic Shortcut
When solving inverse sine problems in exams, remember special sine values: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2. If you memorize sin and corresponding angles, you can instantly answer most inverse sine questions without a calculator. Many Vedantu teachers use mnemonic stories (“Silly Harry Took Oats Away”) to help students remember these values quickly.
Try These Yourself
- Find sin-1(–1/2)
- What is the value of sin-1(0)?
- Calculate the angle whose sine is √3/2
- Which value is outside the domain of inverse sine: 1.2 or –0.8?
- Evaluate sin-1(1)
Frequent Errors and Misunderstandings
- Trying to find sin-1(x) for x values not between –1 and 1
- Giving answers outside –90° to 90° (–π/2 to π/2)
- Confusing the inverse sine with cosecant (which is 1/sin x, not the angle)
Relation to Other Concepts
The idea of inverse sine connects closely with trigonometric functions and other inverse trigonometric functions. Mastering this helps you solve equations, right triangle problems, and even calculus questions involving derivatives and integrals of inverse trig functions. For example, the derivative of sin-1x is 1/√(1–x²).
Classroom Tip
A quick way to remember the domain of inverse sine: only plug in values between –1 and 1. Imagine the sine wave on a graph—these are the only heights reached! Vedantu’s live teaching sessions regularly include interactive demos and practice so you don’t forget such facts before exams.
We explored inverse sine—from definition, formula, value table, mistakes, and connections to other concepts. Keep practicing with Vedantu to get comfortable recognizing standard sine values and quickly working backwards to angles using the sin⁻¹ function.
Explore More: Trigonometric Identities | Trigonometric Values | Sine, Cosine and Tangent Table | Inverse Trigonometric Functions
FAQs on Inverse Sine (sin⁻¹) – Meaning, Formula & Calculation
1. What is the inverse sine (sin⁻¹) function and what does it represent?
The inverse sine function, denoted as sin⁻¹(x) or arcsin(x), is a function that 'undoes' the sine function. It takes a numerical value (between -1 and 1) and gives you the angle whose sine is that value. For example, if sin(30°) = 0.5, then sin⁻¹(0.5) = 30°.
2. What is the principal value branch of the inverse sine function and why is it important?
The principal value branch is the restricted range of the inverse sine function, which is [-π/2, π/2] or [-90°, 90°]. This restriction is crucial because the sine function is periodic (it repeats values). By limiting the output to this specific range, we ensure that the inverse sine function gives a single, unique angle for every input, making it a well-defined function as per mathematical rules.
3. What are the domain and range of y = sin⁻¹x according to the Class 12 NCERT syllabus?
As per the CBSE and NCERT syllabus for Class 12 Maths, the inverse sine function y = sin⁻¹x is defined with the following:
- Domain: The set of all possible input values for x, which is [-1, 1].
- Range (Principal Value Branch): The set of all possible output values for y, which is [-π/2, π/2].
4. How do you find the value of sin⁻¹(x) for common angles like sin⁻¹(1/2) or sin⁻¹(-√3/2)?
To find the value, you ask: "Which angle in the range [-90°, 90°] has a sine of x?"
- For sin⁻¹(1/2), you recall that sin(30°) = 1/2. Since 30° is within the range, sin⁻¹(1/2) = 30° or π/6.
- For sin⁻¹(-√3/2), you use the property sin(-θ) = -sin(θ). We know sin(60°) = √3/2, so sin(-60°) = -√3/2. Since -60° is in the range, sin⁻¹(-√3/2) = -60° or -π/3.
5. What is the crucial difference between inverse sine (sin⁻¹x) and cosecant (csc x)?
This is a common point of confusion. The key difference lies in their purpose:
- Inverse Sine (sin⁻¹x): It is an inverse function that gives an angle as its output. It answers the question, "What angle has this sine value?"
- Cosecant (csc x): It is a reciprocal trigonometric ratio, defined as 1/sin(x). It gives a numerical ratio, not an angle.
In short, sin⁻¹x is about finding an angle, while csc(x) is about finding the reciprocal of a sine value.
6. Why can't you calculate the inverse sine of a number like 2 or -1.5?
You cannot calculate the inverse sine of a value outside the interval [-1, 1] because the original sine function, sin(θ), only produces values within that range. The graph of sin(θ) never goes above 1 or below -1. Since inverse sine's job is to reverse this, its domain is limited to the possible outputs of the sine function, which is [-1, 1].
7. How is the concept of inverse sine used in real-world examples, such as in physics or engineering?
Inverse sine is essential for finding angles when ratios of sides are known. Key applications include:
- Physics: Calculating the angle of a projectile's trajectory, determining the angle of refraction in optics (Snell's Law), and analysing wave forms in simple harmonic motion.
- Engineering: Finding the angle of inclination for ramps, designing stable building structures, and in navigation for determining bearing.
8. What is the derivative of the inverse sine function, d/dx (sin⁻¹x)?
The derivative of the inverse sine function is a fundamental formula in calculus. It is given by:
d/dx (sin⁻¹x) = 1 / √(1 - x²)
This formula is valid for x in the open interval (-1, 1) and is frequently used in integration and solving differential equations.
9. What is the relationship defined by the property sin⁻¹(sin x) = x? Does it always hold true?
The property sin⁻¹(sin x) = x is a common conceptual trap because it only holds true if x lies within the principal value range of the inverse sine function, which is [-π/2, π/2]. If x is outside this range, you must first find an equivalent angle within the range that has the same sine value. For example, sin⁻¹(sin π) is not π, but 0, because sin(π) = 0 and sin⁻¹(0) = 0.
10. How is the graph of y = sin⁻¹x drawn and what are its key features?
The graph of y = sin⁻¹x is essentially a reflection of a segment of the y = sin x graph across the line y = x. Key features are:
- It is defined only on the domain [-1, 1].
- Its range is restricted to [-π/2, π/2].
- The graph is an increasing function that passes through the origin (0,0).
- It starts at the point (-1, -π/2) and ends at the point (1, π/2).
