https://file.aiquickdraw.com/mathbot/file/1726771493768qtu7k0sh.jpg

The triangle in the image provided is a right triangle, with the right angle at \( C \), and sides given as \( CD = 42 \), \( CE = \sqrt{85} \), and hypotenuse \( ED = 43 \).

To solve for the trigonometric ratios, we will first determine the relevant sides for the given trigonometric functions.

### 1. For \( \sin D \):
In a right triangle, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

For angle \( D \):
- The side opposite to \( D \) is \( CE = \sqrt{85} \),
- The hypotenuse is \( ED = 43 \).

Thus:
\[
\sin D = \frac{\sqrt{85}}{43}
\]

### 2. For \( \cos E \):
In a right triangle, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

For angle \( E \):
- The side adjacent to \( E \) is \( CD = 42 \),
- The hypotenuse is \( ED = 43 \).

Thus:
\[
\cos E = \frac{42}{43}
\]

So, the trigonometric ratios are:
- \( \sin D = \frac{\sqrt{85}}{43} \),
- \( \cos E = \frac{42}{43} \).

Would you like further details or clarifications?

### Further questions:
1. How can we derive other trigonometric ratios for \( D \) and \( E \)?
2. What is the formula for \( \cos D \) and \( \sin E \)?
3. How do we calculate the tangent of angle \( D \) or \( E \)?
4. How does the Pythagorean theorem confirm the side lengths of the triangle?
5. What are the key differences between \( \sin \), \( \cos \), and \( \tan \) in terms of side lengths?

### Tip:
In right triangles, remember that the sine and cosine of complementary angles \( \theta \) and \( 90^\circ - \theta \) are related: \( \sin \theta = \cos(90^\circ - \theta) \) and vice versa.

Express the trig ratios as fractions in simplest terms. Given triangle ECD, with sides CD = 42, CE = √85, and hypotenuse ED = 43, find sin D and cos E.

Learn how to find the trigonometric ratios sin D and cos E for triangle ECD, where sides are given as CD = 42, CE = √85, and hypotenuse ED = 43. This problem covers key trigonometric concepts like sine and cosine and applies them to right triangles. Suitable for Grades 9-11.

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