What are sin D

Question

and sec D
?

Right triangle C D E is shown. Side C D is the hypotenuse and it is labeled 41. Side D E is adjacent to angle D and it is labeled 40. Side E C is opposite angle D and it is labeled 9.

Answer 1

In a right triangle, the sine and secant functions can be defined based on the sides of the triangle in relation to one of the angles.

Given triangle CDE:

• CD (hypotenuse) = 41
• DE (adjacent to angle D) = 40
• EC (opposite angle D) = 9

To find \( \sin D \):

The sine of an angle in a right triangle is given by the ratio of the length of the opposite side to the length of the hypotenuse.

\[ \sin D = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{EC}{CD} = \frac{9}{41} \]

To find \( \sec D \):

The secant of an angle is the reciprocal of the cosine of that angle. Cosine is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

First, calculate \( \cos D \): \[ \cos D = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{DE}{CD} = \frac{40}{41} \]

Now, find \( \sec D \): \[ \sec D = \frac{1}{\cos D} = \frac{1}{\frac{40}{41}} = \frac{41}{40} \]

Summary:

• \( \sin D = \frac{9}{41} \)
• \( \sec D = \frac{41}{40} \)