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We are asked to find the length of side \( w \) in a right-angled triangle where:

- The opposite side (to the given angle of \( 60^\circ \)) is \( 1 \) cm.
- The angle is \( 60^\circ \).
- \( w \) is the adjacent side.

We can use the trigonometric relation for tangent since it relates the opposite side and adjacent side:

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

Substituting the known values:

\[
\tan(60^\circ) = \frac{1}{w}
\]

The value of \( \tan(60^\circ) \) is \( \sqrt{3} \). So the equation becomes:

\[
\sqrt{3} = \frac{1}{w}
\]

To solve for \( w \), we rearrange:

\[
w = \frac{1}{\sqrt{3}}
\]

To rationalize the denominator:

\[
w = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

Thus, the length of side \( w \) is \( \frac{\sqrt{3}}{3} \) cm.

Would you like further explanation or details? Here are some related questions:

1. What are the sine and cosine values for a \( 60^\circ \) angle?
2. How would the equation change if the angle was \( 30^\circ \) instead of \( 60^\circ \)?
3. Can you explain the process of rationalizing a denominator?
4. How would you calculate the hypotenuse using the given information?
5. Why is tangent the appropriate trigonometric function in this case?

**Tip:** In any right triangle, the choice of trigonometric function depends on the sides involved—opposite, adjacent, or hypotenuse.

Work out the length of side w. Give your answer in its simplest form.

This problem involves solving for the length of side w in a right-angled triangle where the angle is 60° and the opposite side is 1 cm. Using the tangent function and rationalizing the denominator, the final length of side w is expressed as √3/3 cm. Perfect for students in Grades 8-10.

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