https://file.aiquickdraw.com/mathbot/file/17290301360799xutg5l5.jpg

The image shows a right triangle with two 45° angles, and one 90° angle, which is a 45°-45°-90° triangle. The sides of this triangle follow a specific ratio:

- The legs are equal, each having a length of 1.
- The hypotenuse is \( \sqrt{2} \).

We are asked to find \( \cos(45^\circ) \). In a right triangle, the cosine of an angle is defined as:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

For \( 45^\circ \), one of the legs (which is adjacent) is 1, and the hypotenuse is \( \sqrt{2} \). Thus:

\[
\cos(45^\circ) = \frac{1}{\sqrt{2}}
\]

To match this with the given options:

- A. \( \frac{1}{\sqrt{2}} \) – This is correct.
- B. \( \frac{1}{2} \) – Incorrect.
- C. \( \sqrt{2} \) – Incorrect.
- D. \( 1 \) – Incorrect.

The correct answer is **A: \( \frac{1}{\sqrt{2}} \)**.

Would you like more details on how this triangle works or on trigonometric functions?

Here are some related questions to expand on this:

1. What is the sine of 45° in a 45°-45°-90° triangle?
2. Why is the cosine of 45° equal to \( \frac{1}{\sqrt{2}} \)?
3. How can we rationalize the denominator of \( \frac{1}{\sqrt{2}} \)?
4. What are the side ratios for a 30°-60°-90° triangle?
5. How can we apply the Pythagorean Theorem to verify the hypotenuse as \( \sqrt{2} \)?

**Tip:** When dealing with square roots in denominators, it's often helpful to rationalize the denominator by multiplying both the numerator and denominator by the square root.

This page provides a detailed explanation of how to calculate cos 45° using a 45°-45°-90° triangle. Learn how the sides relate to the cosine function and apply the formula cos(θ) = adjacent / hypotenuse. Suitable for students in Grades 9-10.

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