The image shows a right triangle where the following is given:

• One angle is $41^\circ$
• Another angle is $26^\circ$
• The adjacent side (to the $41^\circ$ angle) is labeled as 520
• We are tasked with finding the opposite side $x$ to the nearest whole number.

Solution:

We can use the tangent function, which relates an angle in a right triangle to the opposite and adjacent sides:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

For the $41^\circ$ angle:

$\tan(41^\circ) = \frac{x}{520}$

Solving for $x$:

$x = 520 \times \tan(41^\circ)$

Now, calculate $\tan(41^\circ)$:

$\tan(41^\circ) \approx 0.8693$

$x = 520 \times 0.8693 \approx 451.636$

Rounding to the nearest whole number:

$x \approx 452$

So, the length of $x$ is approximately 452 units.

Would you like a detailed breakdown of the steps? Here are 5 related questions to expand on this:

1. How do you solve for the adjacent side using trigonometric functions?
2. What would happen if we used the $26^\circ$ angle instead?
3. Can we use the sine or cosine rule in this situation?
4. How do trigonometric functions apply to non-right triangles?
5. What other real-life applications use tangent in similar problems?

Tip: In any right triangle problem, the tangent function is ideal when you have an angle and know the lengths of the opposite and adjacent sides.