Approximate value of x in a right triangle using trigonometric ratios.
Check the final answer first, then review the worked steps.
Problem
Approximate value of x in a right triangle using trigonometric ratios.
Answer
39.59
Step-by-step solution
- Identify the problem type: This is a trigonometry problem involving a right-angled triangle. We need to find the length of a side using trigonometric ratios.
- Analyze the given information: We have a right-angled triangle. One angle is given as $43^\circ$. The side opposite to the $43^\circ$ angle has a length of 27. The side labeled $x$ is the hypotenuse. We are also given approximate values for $\sin 43^\circ$, $\cos 43^\circ$, and $\tan 43^\circ$.
- Choose the appropriate trigonometric ratio: We need to relate the angle ($43^\circ$), the opposite side (27), and the hypotenuse ($x$). The sine function relates these three: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.
- Set up the equation: Using the sine function, we have $\sin(43^\circ) = \frac{27}{x}$.
- Solve for x: To solve for $x$, we can rearrange the equation: $x = \frac{27}{\sin(43^\circ)}$.
- Substitute the given value for sin(43°): From the table, we know that $\sin 43^\circ \approx 0.682$. So, $x \approx \frac{27}{0.682}$.
- Calculate the value of x: $x \approx 39.58947...$
- Round to the nearest hundredth: Rounding the result to two decimal places, we get $x \approx 39.59$.