Solve Without A Calculator Cos X Sqrt 3 3

Introduction

The equation cos x = √3/3 is a fundamental trigonometric equation that appears in many mathematical and scientific contexts. Solving it without a calculator involves understanding the unit circle and reference angles.

This guide covers three primary methods to solve cos x = √3/3:

1. Using reference angles and unit circle values
2. Using the inverse cosine function
3. Using the general solution formula

Methods to Solve cos x = √3/3

Method 1: Using Reference Angles and Unit Circle Values

First, recognize that √3/3 is equal to sin(π/6) and cos(π/6). This is because:

However, we need to find x where cos x = √3/3. The exact value is cos(π/6 + 2πn) or cos(5π/6 + 2πn) for any integer n, because:

But we need cos x = √3/3, not √3/2. This suggests we need to find an angle whose cosine is √3/3.

Method 2: Using the Inverse Cosine Function

The primary solution can be found using the inverse cosine function:

Where n is any integer. The approximate value of cos⁻¹(√3/3) is about 0.9273 radians (53.13°).

Method 3: Using the General Solution Formula

The general solution for cos x = a (where -1 ≤ a ≤ 1) is:

For our equation, this becomes:

This formula gives all solutions in radians for any integer n.

Worked Examples

Example 1: Finding Principal Solutions

Find the principal solutions (smallest positive and negative solutions) for cos x = √3/3.

Using the inverse cosine function:

Example 2: Finding Solutions in a Specific Range

Find all solutions for cos x = √3/3 in the interval [0, 2π).

Using the general solution formula:

These are the only two solutions in the interval [0, 2π).

Formula Explanation

The general solution for cos x = a is:

Where:

• a is the value of the cosine function (must be between -1 and 1)
• cos⁻¹(a) is the inverse cosine function, giving the principal value in radians
• n is any integer (0, ±1, ±2, etc.)

For cos x = √3/3, we substitute a = √3/3:

This formula gives all possible solutions for the equation.