What is General Solution of tan x = k?

S6-SA2-0333

Grade Level:

Class 10

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

Definition

What is it?

The General Solution of tan x = k is a formula that gives ALL possible angles 'x' for which the tangent of 'x' equals a specific number 'k'. It helps us find every angle that satisfies the equation, not just one.

Simple Example

Quick Example

Imagine you know tan x = 1. You might quickly think x = 45 degrees. But what about 225 degrees? Or 405 degrees? The general solution is like a special 'formula' that helps you find ALL these angles (45, 225, 405, etc.) without missing any, just like a bus route tells you all the stops.

Worked Example

Step-by-Step

Let's find the general solution for tan x = sqrt(3).

Step 1: First, find the principal value (the smallest positive angle) for which tan x = sqrt(3). We know that tan 60 degrees = sqrt(3).

---Step 2: Convert this angle to radians. 60 degrees = pi/3 radians.

---Step 3: Now, use the general solution formula for tan x = tan alpha, which is x = n*pi + alpha, where 'n' is any integer (like -2, -1, 0, 1, 2, ...).

---Step 4: Substitute alpha = pi/3 into the formula.

---Step 5: So, x = n*pi + pi/3.

---Step 6: We can also write this as x = (3n + 1)*pi / 3.

Answer: The general solution for tan x = sqrt(3) is x = n*pi + pi/3, where n is an integer.

Why It Matters

Understanding general solutions helps engineers design stable structures and analyze vibrations in machines. It's crucial for physicists calculating wave patterns and for computer scientists developing graphics and animations in games. This concept is vital for careers in engineering, physics, and even AI/ML.

Common Mistakes

MISTAKE: Forgetting to add 'n*pi' in the general solution | CORRECTION: Always remember that the tangent function repeats every pi radians (or 180 degrees), so 'n*pi' is essential to capture all solutions.

MISTAKE: Confusing the general solution for tan x = k with those for sin x = k or cos x = k | CORRECTION: Each trigonometric function has a unique general solution formula. For tan x = k, it's x = n*pi + alpha, where alpha is the principal value.

MISTAKE: Using degrees instead of radians in the general solution formula when the problem expects radians | CORRECTION: Always check the units required. The formula x = n*pi + alpha typically assumes alpha is in radians. Convert degrees to radians (e.g., 180 degrees = pi radians) before applying the formula.

Practice Questions

Try It Yourself

QUESTION: Find the general solution for tan x = 1. | ANSWER: x = n*pi + pi/4, where n is an integer.

QUESTION: If tan x = -1/sqrt(3), find the general solution. (Hint: The principal value for tan x = -1/sqrt(3) is -pi/6 or 11pi/6). | ANSWER: x = n*pi - pi/6, where n is an integer.

QUESTION: A pendulum swings such that its angle 'x' (in radians) from the vertical satisfies tan x = 0. Find all possible values of 'x'. | ANSWER: x = n*pi, where n is an integer.

MCQ

Quick Quiz

What is the general solution for tan x = 0?

x = 2n*pi

x = n*pi + pi/2

x = n*pi

x = n*pi - pi/2

The Correct Answer Is:

For tan x = 0, the principal value is 0 (since tan 0 = 0). Using the general solution formula x = n*pi + alpha, we get x = n*pi + 0, which simplifies to x = n*pi.

Real World Connection

In the Real World

Imagine you're an engineer designing a satellite for ISRO. You need to calculate the exact angles for its solar panels to always face the sun. The sun's position changes, and trigonometric equations like tan x = k help find these angles. General solutions ensure the panels are positioned correctly at all times, making sure the satellite gets enough power.

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 | RADIANS: A unit of angle measurement, where pi radians equals 180 degrees | INTEGER: A whole number (positive, negative, or zero) | TANGENT FUNCTION: A trigonometric ratio defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

What's Next

What to Learn Next

Great job understanding general solutions for tan x = k! Next, you should explore the 'General Solutions for sin x = k and cos x = k'. These build on similar ideas but have slightly different formulas, which are super important for solving more complex trigonometry problems in Class 11 and 12.