To find the missing side $c$ of the right triangle, we can use the trigonometric relationship:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Here:

• $\theta = 66.8^\circ$
• The opposite side = 17.4 cm
• Hypotenuse ($c$) = the missing quantity

Rewriting the formula: $c = \frac{\text{opposite}}{\sin(\theta)}$

Substitute the values: $c = \frac{17.4}{\sin(66.8^\circ)}$

Now calculate:

1. Find $\sin(66.8^\circ)$ using a calculator or trigonometric table. $\sin(66.8^\circ) \approx 0.9175$
2. Compute $c$: $c = \frac{17.4}{0.9175} \approx 18.96 \, \text{cm}$

Round to the nearest tenth: $c \approx 19.0 \, \text{cm}$

Final Answer:

$c \approx 19.0 \, \text{cm}$

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Do you want more details on the steps or have any questions?
Here are five related questions for further exploration:

1. What is the relationship between the sine, cosine, and tangent functions in a right triangle?
2. How can you use the Pythagorean theorem to verify your result?
3. How would you solve for the angle if only the two sides were given?
4. How does the choice of trigonometric ratio change if you are given adjacent instead of opposite?
5. What is the difference between using degrees and radians in trigonometry?

Tip: Always ensure your calculator is set to the correct angle mode (degrees or radians) to