In which triangle is the value of $x$ equal to $\cos ^{-1}\left(\frac{4.3}{6.7}\right) ?
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Answer (1)
The problem states that cos ( x ) = 6.7 4.3 .
Triangle 1 has adjacent side 4.3 and hypotenuse 6.7, so cos ( x ) = 6.7 4.3 .
Triangle 2 has adjacent side 6.7 and hypotenuse 4.3, so cos ( x ) = 4.3 6.7 .
Triangle 3 has adjacent side 4.3 and hypotenuse 7.5, so cos ( x ) = 7.5 4.3 .
Therefore, the value of x is equal to cos − 1 ( 6.7 4.3 ) in Triangle 1 .
Explanation
Analyze the problem We are given that x = cos − 1 ( 6.7 4.3 ) . This means that cos ( x ) = 6.7 4.3 . We need to find the triangle where the ratio of the adjacent side to the hypotenuse is equal to 6.7 4.3 ≈ 0.6418 .
Examine each triangle Let's examine each triangle:
Triangle 1: The adjacent side to angle x is 4.3, and the hypotenuse is 6.7. Therefore, cos ( x ) = 6.7 4.3 ≈ 0.6418 .
Triangle 2: The adjacent side to angle x is 6.7, and the hypotenuse is 4.3. Therefore, cos ( x ) = 4.3 6.7 ≈ 1.5581 .
Triangle 3: The adjacent side to angle x is 4.3, and the hypotenuse is 7.5. Therefore, $\cos(x) = \frac{4.3}{7.5} \approx 0.5733.
Compare the ratios Comparing the calculated ratios with the given value of cos ( x ) = 6.7 4.3 ≈ 0.6418 , we can see that Triangle 1 has the correct ratio.
Final Answer Therefore, the value of x is equal to cos − 1 ( 6.7 4.3 ) in Triangle 1.
Examples
Understanding trigonometric functions like cosine is crucial in various fields. For example, architects use cosine to calculate roof angles, ensuring proper water runoff. Similarly, engineers apply cosine when designing bridges to determine the forces acting on different structural components. In navigation, cosine helps calculate distances and bearings, allowing ships and airplanes to follow precise routes. These applications demonstrate how a simple concept like cosine plays a vital role in real-world scenarios.
