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To find the length of \( z \), we can apply trigonometric ratios based on the information provided. In this right triangle, we know:

- One angle is \( 61^\circ \).
- The adjacent side to the \( 61^\circ \) angle is 10 cm.
- The value of \( \cos(61^\circ) = 0.5 \).
- The value of \( \sin(61^\circ) = 0.9 \).

Since \( z \) is the hypotenuse and we know the adjacent side (10 cm), we can use the cosine ratio:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

\[
0.5 = \frac{10}{z}
\]

Now, solve for \( z \):

\[
z = \frac{10}{0.5} = 20 \, \text{cm}
\]

Thus, the length of \( z \) is 20 cm.

Would you like further explanation, or have any other questions?

Here are 5 related questions to consider:

1. What trigonometric ratio would we use if we were given the opposite side instead of the adjacent side?
2. How can we verify the value of \( z \) using the sine function?
3. Can you solve for the length of the opposite side using the sine of 61°?
4. How would the calculation change if the given angle was 29° instead of 61°?
5. What other methods could we use to solve for \( z \) if trigonometric functions were not available?

**Tip**: When working with trigonometry, always double-check which side corresponds to the angle to choose the correct ratio (sine, cosine, or tangent).

Find z in the right triangle where one angle is 61°, the adjacent side to the 61° angle is 10 cm, the cosine of 61° is 0.5, and the sine of 61° is 0.9.

Learn how to find the hypotenuse (z) of a right triangle where the adjacent side is 10 cm, one angle is 61°, and the cosine of 61° is 0.5. The solution involves using basic trigonometric ratios. Ideal for students in Grades 9-11.

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