Let's solve the equation \( 0 = 166.88 \times \tan(33^\circ 52' 30'') + 1.60 - 2.043 \).

### Step 1: Convert the Angle to Decimal Form
First, convert the angle \( 33^\circ 52' 30'' \) into decimal degrees:

\[
\text{Decimal degrees} = 33 + \frac{52}{60} + \frac{30}{3600} = 33.875^\circ
\]

### Step 2: Calculate the Tangent
Now, compute the tangent of the angle \( 33.875^\circ \):

\[
\tan(33.875^\circ) \approx 0.66913
\]

### Step 3: Multiply by the Given Constant
Next, multiply the tangent value by 166.88:

\[
166.88 \times 0.66913 \approx 111.618
\]

### Step 4: Substitute and Solve the Equation
Substitute this value into the equation and simplify:

\[
0 = 111.618 + 1.60 - 2.043
\]

Simplify further:

\[
0 = 111.175
\]

This equation does not hold, which indicates that the equation has no real solution.

Would you like more details or have any questions?

Here are some related questions:
1. What happens if you change the angle slightly in the equation?
2. How would the solution change if you used a different trigonometric function?
3. Can you express the angle in radians instead of degrees and solve it?
4. How do you find the tangent of an angle without a calculator?
5. What does it mean when an equation like this has no solution?

**Tip:** Always ensure that the angle is in the correct units (degrees vs. radians) before performing trigonometric calculations.

0 = 166.88 * tg 33° 52' 30" + 1.60 - 2.043

Learn how to solve the trigonometric equation 0 = 166.88 * tan(33° 52' 30") + 1.60 - 2.043 by converting angles to decimal degrees and using tangent calculations. This solution is ideal for students in Grades 10-12 interested in exploring trigonometry and algebraic simplifications.

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