What is General Solution of sin x = k?

S6-SA2-0331

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The general solution of sin x = k is a formula that gives ALL possible angles 'x' for which the sine of that angle equals a specific value 'k'. Since the sine function repeats its values after every 360 degrees (or 2*pi radians), there are infinitely many solutions, and the general solution captures them all.

Simple Example

Quick Example

Imagine you are watching a Ferris wheel at a funfair. If the height of a specific seat above the ground can be described by a sine function, then finding 'x' when sin x = k is like finding all the different times the seat will be at a particular height 'k'. Since the wheel keeps spinning, it will reach that height many times.

Worked Example

Step-by-Step

Let's find the general solution for sin x = 1/2.
---1. First, find the principal value. We know that sin(30 degrees) = 1/2. In radians, this is sin(pi/6) = 1/2.
---2. Here, the value of k is 1/2, which is between -1 and 1. So, a solution exists.
---3. The principal value (alpha) is pi/6.
---4. The general solution for sin x = k is given by the formula: x = n*pi + (-1)^n * alpha, where 'n' is any integer (0, 1, -1, 2, -2, ...).
---5. Substitute alpha = pi/6 into the formula.
---6. So, the general solution is x = n*pi + (-1)^n * (pi/6).
Answer: x = n*pi + (-1)^n * (pi/6), where n belongs to integers (Z).

Why It Matters

Understanding general solutions helps engineers design systems that repeat, like the vibrations in a bridge or the rhythm of a heart monitor in medicine. In space technology, it helps predict the repeating orbits of satellites. This knowledge is crucial for careers in engineering, physics, and even AI/ML for analyzing wave patterns.

Common Mistakes

MISTAKE: Only giving one or two solutions (e.g., 30 degrees and 150 degrees) instead of the general formula. | CORRECTION: Remember that trigonometric functions are periodic, so there are infinite solutions. Always use the general formula to express all possible values.

MISTAKE: Confusing the general solution for sin x = k with that for cos x = k or tan x = k. | CORRECTION: Each trigonometric function has its own unique general solution formula. Make sure to use the correct formula for 'sin x = k'.

MISTAKE: Forgetting that 'n' in the formula can be any integer (positive, negative, or zero). | CORRECTION: Always state that 'n' belongs to the set of integers (Z) to include all possible cycles of the function.

Practice Questions

Try It Yourself

QUESTION: Find the general solution for sin x = 0. | ANSWER: x = n*pi, where n belongs to integers (Z).

QUESTION: Find the general solution for sin x = 1. | ANSWER: x = n*pi + (-1)^n * (pi/2), or simplified as x = (2n + 1/2)*pi, where n belongs to integers (Z).

QUESTION: If sin x = -sqrt(3)/2, find the general solution. | ANSWER: x = n*pi + (-1)^n * (-pi/3), or x = n*pi - (-1)^n * (pi/3), where n belongs to integers (Z). (Note: principal value could be -pi/3 or 4pi/3, both lead to equivalent general solutions)

MCQ

Quick Quiz

What is the general solution for sin x = k, where alpha is the principal value?

x = n*pi + alpha

x = 2n*pi +/- alpha

x = n*pi + (-1)^n * alpha

x = n*pi - alpha

The Correct Answer Is:

The correct general solution for sin x = k is x = n*pi + (-1)^n * alpha. Options A, B, and D are incorrect formulas for the sine function's general solution.

Real World Connection

In the Real World

In India, engineers at ISRO use principles like general solutions to calculate the precise, repeating trajectories of satellites orbiting Earth. They need to know all the possible times a satellite will be at a specific point in its orbit to plan communication windows or course corrections, ensuring our mobile network and weather forecasts work smoothly.

Key Vocabulary

Key Terms

GENERAL SOLUTION: A formula that gives all possible values for a variable in a trigonometric equation. | PRINCIPAL VALUE: The smallest positive or negative angle that satisfies a trigonometric equation. | PERIODIC FUNCTION: A function that repeats its values in regular intervals. | INTEGER: Any whole number (positive, negative, or zero). | RADIANS: A unit for measuring angles, where 180 degrees = pi radians.

What's Next

What to Learn Next

Now that you understand the general solution for sin x = k, you should explore the general solutions for cos x = k and tan x = k. These build directly on what you've learned and will complete your understanding of solving basic trigonometric equations.