In which triangle is the value of [tex]$x$[/tex] equal to [tex]$\cos ^{-1}\left(\frac{4.3}{6.7}\right)$[/tex]?
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Answer (2)
The value of x is equal to cos − 1 ( 6.7 4.3 ) in Triangle B, as it is the only triangle where cos ( x ) = 6.7 4.3 .
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The problem states that x = cos − 1 ( 6.7 4.3 ) , implying cos ( x ) = 6.7 4.3 .
Examine each triangle and calculate the cosine of angle x in each case.
In triangle (A), cos ( x ) = 4.3 6.7 . In triangle (B), cos ( x ) = 6.7 4.3 . In triangle (C), cos ( x ) = 7.8 6.7 .
Triangle (B) satisfies the condition cos ( x ) = 6.7 4.3 . Therefore, the answer is triangle (B).
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 in which the ratio of the adjacent side to the hypotenuse with respect to angle x is equal to 6.7 4.3 .
Examine each triangle From the python calculation tool, we know that 6.7 4.3 ≈ 0.6418 . Now, let's examine each triangle to see which one satisfies the condition cos ( x ) = 6.7 4.3 .
In triangle (A), the side adjacent to angle x is 6.7, and the hypotenuse is 4.3. Therefore, cos ( x ) = 4.3 6.7 , which is not equal to 6.7 4.3 .
In triangle (B), the side adjacent to angle x is 4.3, and the hypotenuse is 6.7. Therefore, cos ( x ) = 6.7 4.3 , which matches the given condition.
In triangle (C), the side adjacent to angle x is 6.7, and the hypotenuse is 7.8. Therefore, cos ( x ) = 7.8 6.7 , which is not equal to 6.7 4.3 .
Final Answer Therefore, the value of x is equal to cos − 1 ( 6.7 4.3 ) in triangle (B).
Examples
Understanding trigonometric ratios like cosine is crucial in various real-world applications. For instance, when designing a ramp for accessibility, the angle of the ramp needs to be carefully calculated to meet safety standards. The cosine of this angle, determined by the ratio of the ramp's horizontal length to its actual length, helps ensure the ramp is neither too steep nor too long, providing a safe and manageable incline for wheelchair users.
