What is General Solution of cos x = k?

S6-SA2-0332

Grade Level:

Class 10

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

Definition

What is it?

The general solution for cos x = k is a formula that gives ALL possible values of 'x' for which the cosine of 'x' equals a specific number 'k'. It helps us find every angle, positive or negative, that satisfies the equation, not just the ones in a single rotation.

Simple Example

Quick Example

Imagine you are watching a Ferris wheel rotate. If 'k' represents the height of a specific seat at a certain time (related to cosine), the general solution tells you all the different times (angles 'x') when that seat will be at exactly that same height, even after many rotations, forwards or backwards.

Worked Example

Step-by-Step

Let's find the general solution for cos x = 1/2.

Step 1: First, find the principal value. We know that cos(pi/3) = 1/2. So, alpha = pi/3.
---Step 2: Check the range of 'k'. Here, k = 1/2, which is between -1 and 1, so a solution exists.
---Step 3: Apply the general solution formula for cos x = cos alpha, which is x = 2n*pi +/- alpha, where 'n' is any integer.
---Step 4: Substitute the value of alpha. So, x = 2n*pi +/- pi/3.
---Answer: The general solution for cos x = 1/2 is x = 2n*pi +/- pi/3, where n belongs to integers (n is an element of Z).

Why It Matters

Understanding general solutions is crucial in fields like Physics for analyzing wave patterns, in Engineering for designing oscillating systems, and in Space Technology for calculating satellite orbits. Engineers and scientists use these solutions to predict and understand periodic phenomena.

Common Mistakes

MISTAKE: Forgetting the '2n*pi' part and only writing x = +/- alpha | CORRECTION: The '2n*pi' term accounts for all full rotations, so it must always be included to find ALL possible solutions.

MISTAKE: Not checking if 'k' is between -1 and 1. For example, trying to find a solution for cos x = 2 | CORRECTION: The cosine function's value can only be between -1 and 1. If 'k' is outside this range, there is no real solution.

MISTAKE: Confusing the general solution for cos x = k with sin x = k or tan x = k | CORRECTION: Each trigonometric function has its own unique general solution formula. For cos x = k, it's x = 2n*pi +/- alpha.

Practice Questions

Try It Yourself

QUESTION: Find the general solution for cos x = sqrt(3)/2. | ANSWER: x = 2n*pi +/- pi/6, where n is an integer.

QUESTION: Find the general solution for cos x = -1/2. | ANSWER: x = 2n*pi +/- 2*pi/3, where n is an integer.

QUESTION: If cos x = 0, find the general solution. | ANSWER: x = (2n+1)*pi/2, where n is an integer. (Hint: cos(pi/2) = 0)

MCQ

Quick Quiz

What is the general solution for cos x = 1?

x = n*pi

x = 2n*pi

x = n*pi/2

x = 2n*pi +/- pi/2

The Correct Answer Is:

For cos x = 1, the principal value is 0 (since cos(0) = 1). Using the formula x = 2n*pi +/- alpha, we get x = 2n*pi +/- 0, which simplifies to x = 2n*pi.

Real World Connection

In the Real World

In India, ISRO scientists use these trigonometric solutions to calculate the precise orbital paths of satellites like Chandrayaan and Mangalyaan. They need to know when a satellite will be at a specific position relative to Earth, which involves solving equations like cos x = k to predict future locations accurately.

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 angle that satisfies a trigonometric equation. | TRIGONOMETRIC EQUATION: An equation involving trigonometric functions of an unknown angle. | INTEGER: Any whole number (positive, negative, or zero).

What's Next

What to Learn Next

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