Solving Trigonometric Equations with Fractions: Detailed Steps and Calculation

Introduction to Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions such as sine, cosine, and tangent. When dealing with fractions in these equations, it's essential to follow a systematic approach to simplify and solve them effectively.

Solving the Given Trigonometric Equation

The problem at hand is:

frac{tan theta 3sin theta 2}{tan theta - 3sin theta 1} 2

Step 1: Cross-Multiplication

To eliminate the fraction, we cross-multiply:

tan theta 3sin theta 2 2tan theta - 6sin theta 2

Step 2: Simplifying the Equation

Next, we simplify the equation by moving all terms to one side:

-tan theta 9sin theta 0

This can be rearranged to:

tan theta 9sin theta

Step 3: Using Trigonometric Identity

Recall the trigonometric identity:

tan theta frac{sin theta}{cos theta}

Substitute this identity into the equation:

frac{sin theta}{cos theta} 9sin theta

Assuming sin theta neq 0, we can divide both sides by sin theta:

frac{1}{cos theta} 9

This leads to:

cos theta frac{1}{9}

Step 4: Finding the Angle

To find the angle theta, we use the inverse cosine function:

theta cos^{-1}left(frac{1}{9}right)

Using a calculator, we find:

theta approx 83.62°

Summary

The solution to the equation:

frac{tan theta 3sin theta 2}{tan theta - 3sin theta 1} 2

for 0° leq theta leq 90° is approximately:

theta approx 83.62°

Additional Steps for Similar Problems

Step 1: Simplify the Expression

If the given expression is:

[tan theta 3sin theta 2]/[tan theta - 3sin theta 1] 2

First, we simplify by cross-multiplying:

[sin theta 3sin theta cos theta 2cos theta]/[sin theta - 3sin theta cos theta cos theta] 2

And further simplifying:

[sin theta 3sin theta cos theta 2cos theta] / [sin theta - 3sin theta cos theta cos theta] 2

Step 2: Combining Like Terms

Using the trigonometric identity:

sin theta 3sin theta cos theta 2cos theta 2sin theta - 6sin theta cos theta

Further simplification gives:

sin theta - 9sin theta cos theta 0

Step 3: Isolating Cosine

At this step, we can isolate cosine:

1 - 9cos theta sin theta 0

Which simplifies to: