Answer :
"The correct equation to find the value of x is:
[tex]\[ \cos 49^\circ = \frac{x}{55} \][/tex]
Given that triangle DEF is a right triangle with angle E as the right angle, we can use trigonometric ratios to find the unknown side x, which is DE. The side DF, which is opposite to angle F, is given as 55 units. Angle F is given as 49 degrees.
The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. In this case, the adjacent side to angle F is DE (which we are trying to find), and the hypotenuse is DF. Therefore, the cosine of angle F (49 degrees) can be expressed as:
[tex]\[ \cos F = \frac{\text{Adjacent side (DE)}}{\text{Hypotenuse (DF)}} \][/tex]
Substituting the known values, we get:
[tex]\[ \cos 49^\circ = \frac{x}{55} \][/tex]
This is the correct equation to solve for x. The other equations provided in the question are incorrect because they either incorrectly place x as the denominator when it should be the numerator or they use the sine function instead of the cosine function. The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse, which is not applicable here for finding the adjacent side DE."
Final answer:
To find the value of x, the adjacent side in triangle DEF, the correct trigonometric equation is cos 49° = x over 55, as it involves the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Explanation:
The question involves finding the value of x in a right triangle DEF, where angle E is a right angle, DE is the adjacent side measuring x, DF is the hypotenuse measuring 55, and angle F measures 49 degrees. To find the value of x, we need to use trigonometric functions. In this case, since we are given the hypotenuse and an angle, and we need to find the adjacent side, the cosine function is most appropriate.
Therefore, the correct equation to find the value of x is cos 49° = x over 55. This is because the cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. The equation can be rewritten and solved to find x, demonstrating the application of trigonometric principles in solving for unknown sides in right triangles.
