In which triangle is the value of $x$ equal to $\cos ^{-1}\left(\frac{4.3}{6.7}\right) ?
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6.7
4.3
Answer (2)
The problem requires finding the triangle where x = cos − 1 ( 6.7 4.3 ) .
Recall that cosine is the ratio of the adjacent side to the hypotenuse.
Identify the adjacent side (4.3) and hypotenuse (6.7) in the given triangle.
Verify that cos ( x ) = 6.7 4.3 , thus x = cos − 1 ( 6.7 4.3 ) .
Conclude that the given triangle is the correct one: $\boxed{\text{The given triangle}}.
Explanation
Analyze the problem The problem asks us to identify the triangle in which the value of x is equal to cos − 1 ( 6.7 4.3 ) . This means we are looking for a right triangle where the ratio of the adjacent side to the hypotenuse is 6.7 4.3 .
Recall the definition of cosine In the given triangle, we need to check if the cosine of angle x is equal to 6.7 4.3 . The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
Identify adjacent and hypotenuse In the given triangle, the side adjacent to angle x has a length of 4.3, and the hypotenuse has a length of 6.7. Therefore, cos ( x ) = 6.7 4.3 .
Conclude the solution Since cos ( x ) = 6.7 4.3 , it follows that x = cos − 1 ( 6.7 4.3 ) . Thus, the given triangle is the one where the value of x is equal to cos − 1 ( 6.7 4.3 ) .
Examples
Understanding trigonometric functions like cosine is crucial in various real-world applications. For example, when designing a ramp for wheelchair access, you need to determine the angle of inclination to meet accessibility standards. If the horizontal distance the ramp covers is 4.3 meters and the ramp's length is 6.7 meters, you can use the cosine function to find the angle: cos ( x ) = 6.7 4.3 , where x is the angle of inclination. This ensures the ramp is safe and compliant.
The angle x is equal to cos − 1 ( 6.7 4.3 ) in a right triangle where 4.3 is the length of the adjacent side and 6.7 is the length of the hypotenuse. Therefore, in the specified triangle, x represents the angle consistent with this ratio. This clearly identifies the triangle in question.
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