Solving for the Length of a Side in a Right-Angled Triangle Using Trigonometry
The image shows a right-angled triangle with sides labeled, and we need to solve for side \( x \). This can be done using trigonometry. The triangle has one side of length 77, which is the opposite side to the angle marked as 68 degrees, and side \( x \) is the hypotenuse.
To find side \( x \), we can use the sine (sin) function, which is defined as the ratio of the length of the opposite side over the hypotenuse in a right-angled triangle:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, \( \theta \) is 68 degrees, the opposite side is 77, and the hypotenuse is \( x \). Plugging these into the equation:
\[ \sin(68^\circ) = \frac{77}{x} \]
We now want to solve for \( x \):
\[ x = \frac{77}{\sin(68^\circ)} \]
Using a calculator:
\[ x \approx \frac{77}{0.927} \]
\[ x \approx \frac{77}{0.927} \]
\[ x \approx 83.1 \]
Thus, the length of side \( x \) is approximately 83.1, to the nearest tenth.
