To find the sine and secant of angle \( D \) in right triangle \( CDE \), we use the definitions of sine and secant:
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The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse: \[ \sin D = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{E C}{C D} \]
From the information provided:
- The length of side \( E C \) (opposite) is \( 9 \).
- The length of side \( C D \) (hypotenuse) is \( 41 \).
Therefore, \[ \sin D = \frac{9}{41} \]
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The secant of an angle is the reciprocal of the cosine of the angle. The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse: \[ \cos D = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{D E}{C D} \]
From the information provided:
- The length of side \( D E \) (adjacent) is \( 40 \).
- The length of side \( C D \) (hypotenuse) is \( 41 \).
Therefore, \[ \cos D = \frac{40}{41} \]
Consequently, the secant of angle \( D \) is given by: \[ \sec D = \frac{1}{\cos D} = \frac{1}{\frac{40}{41}} = \frac{41}{40} \]
In summary:
- \(\sin D = \frac{9}{41}\)
- \(\sec D = \frac{41}{40}\)