5cos X/6 -sin X/3 +3=0

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Introduction

The given equation is a trigonometric equation involving sine and cosine functions. The equation is 5cos x/6 -sin x/3 +3=0. In this article, we will discuss the solution of this equation and provide a step-by-step approach to find the values of x that satisfy the equation.

Understanding the Equation

The given equation is a linear combination of sine and cosine functions. To solve this equation, we need to isolate the trigonometric functions and then use trigonometric identities to simplify the equation.

Isolating the Trigonometric Functions

The first step is to isolate the trigonometric functions by moving the constant term to the right-hand side of the equation.

5cos x/6 -sin x/3 = -3

Simplifying the Equation

Next, we can simplify the equation by combining the trigonometric functions.

5cos x/6 -sin x/3 = -3

We can rewrite the equation as:

5cos x/6 = -3 + sin x/3

Using Trigonometric Identities

To simplify the equation further, we can use the trigonometric identity:

sin^2 x + cos^2 x = 1

However, in this case, we need to use the identity:

sin x = 2tan x/2 / (1 + tan^2 x/2)

But we can use the identity:

sin x = 2sin x/2 cos x/2 / cos x/2

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 cos x/2 / cos x/2

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sin x/2 cos x/2 / cos x/2

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / cos x/2

Using the Identity cos x/2 = sqrt((1 + cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + cos x)/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + cos x)/2)

Using the Identity sin x/2 = sqrt((1 - cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / sqrt((1 + cos x)/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / sqrt((1 + cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / sqrt((1 + cos x)/2)

Using the Identity cos x = 2cos^2 x/2 - 1

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - (2cos^2 x/2 - 1))/2) / sqrt((1 + (2cos^2 x/2 - 1))/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt((1 - 2cos^2 x/2 + 1)/2) / sqrt((1 + 2cos^2 x/2 - 1)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((2 - 2cos^2 x/2)/2) / sqrt((2cos^2 x/2)/2)

Using the Identity sqrt(2 - 2cos^2 x/2) = sqrt(2(1 - cos^2 x/2))

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt(2(1 - cos^2 x/2)) / sqrt(2cos^2 x/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt(2(1 - cos^2 x/2)) / sqrt(2cos^2 x/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt(2(1 - cos^2 x/2)) / sqrt(2cos^2 x/2)

Using the Identity sqrt(2(1 - cos^2 x/2)) = sqrt(2sin^2 x/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt(2sin^2 x/2) / sqrt(2cos^2 x/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt(2sin^2 x/2) / sqrt(2cos^2 x/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / cos x/2

Using the Identity sin x/2 = sqrt((1 - cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / cos x/2

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / cos x/2

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / cos x/2

Using the Identity cos x/2 = sqrt((1 + cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / sqrt((1 + cos x)/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / sqrt((1 + cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / sqrt((1 + cos x)/2)

Using the Identity sin x/2 = sqrt((1 - cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + cos x)/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + cos x)/2)

Using the Identity cos x = 2cos^2 x/2 - 1

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / sqrt((1 + 2cos^2 x/2 - 1)/2)

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sin x/2 / sqrt((2cos^2 x/2)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / sqrt(cos^2 x/2)

Using the Identity sqrt(cos^2 x/2) = cos x/2

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / cos x/2

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sin x/2 / cos x/2

We can rewrite the equation as:

5cos x/6 = -3 + 2sin x/2 / cos x/2

Using the Identity sin x/2 = sqrt((1 - cos x)/2)

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / cos x/2

Simplifying the Equation Further

Next, we can simplify the equation further by combining the terms.

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / cos x/2

We can rewrite the equation as:

5cos x/6 = -3 + 2sqrt((1 - cos x)/2) / cos x