Question - Solving for the Length of a Side in a Right-Angled Triangle Using Trigonometry
Solution:
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.
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