Draw a right-angled triangle containing an angle of 45°. If one of the shorter sides has length one, write down the lengths of the other sides without using trigonometry.
Hence find the exact values of sin 45°, cos 45° and tan 45°.

To solve this problem, we start by drawing a right-angled triangle with one angle of 45°. In a right-angled triangle, the two non-right angles must sum to 90°. Therefore, if one of these angles is 45°, the other must also be 45°. This makes the triangle an isosceles right-angled triangle, where the two legs are of equal length.

Given that one of the shorter sides (one leg of the triangle) has a length of 1, we can determine the lengths of the other sides as follows:

1. **Both legs are equal in length:** Since the triangle is isosceles and one leg is 1, the other leg is also 1.

2. **Hypotenuse calculation:** The length of the hypotenuse (let's denote it as \( c \)) can be found using the Pythagorean theorem, which states:

   \[
   c^2 = a^2 + b^2
   \]

   Substituting the known lengths:

   \[
   c^2 = 1^2 + 1^2 = 1 + 1 = 2
   \]

   Thus, \( c = \sqrt{2} \).

Now, we can find the exact values of the trigonometric functions for the 45° angle:

1. **\(\sin 45^\circ\)**:
   \[
   \sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
   \]

2. **\(\cos 45^\circ\)**:
   \[
   \cos 45^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
   \]

3. **\(\tan 45^\circ\)**:
   \[
   \tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1
   \]

For exact values, we often rationalize the denominators:

- \(\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
- \(\tan 45^\circ = 1\)

Thus, the exact values are:

- \(\sin 45^\circ = \frac{\sqrt{2}}{2}\)
- \(\cos 45^\circ = \frac{\sqrt{2}}{2}\)
- \(\tan 45^\circ = 1\)

Would you like any further explanation or have any questions about these results?

Here are 5 questions you might consider asking next:
1. How can we use trigonometry to find the area of the triangle?
2. What are the properties of isosceles right-angled triangles?
3. Can you explain why the sine and cosine of 45° are equal?
4. How do you rationalize a fraction with a square root in the denominator?
5. What are some real-life applications of right-angled triangles?

**Tip:** In trigonometry, rationalizing denominators is often done to simplify expressions, making them easier to work with or recognize.

Discover the exact values of sin 45°, cos 45°, and tan 45° using a right-angled triangle with a 45° angle. Explore how the lengths of sides relate to trigonometric functions without using trigonometry directly. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation