Trigonometry 45 Degree Triangle

45-45-90 Triangle Properties:

\[ \sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2} \]

In a 45-45-90 triangle, the legs are equal and the hypotenuse is \( \sqrt{2} \) times the length of either leg.

Hypotenuse:

Area:

Perimeter:

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1. What is a 45-45-90 Triangle?

Definition: A special right triangle where both acute angles are 45 degrees, making it isosceles with two equal sides.

Properties: Both legs are equal, and the hypotenuse is always \( \sqrt{2} \) times the length of either leg.

2. Key Trigonometric Values

For a 45 degree angle:

\[ \sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2} \approx 0.7071 \] \[ \tan(45°) = 1 \]

3. Formulas for 45-45-90 Triangle

Hypotenuse: \( h = l \times \sqrt{2} \)
Area: \( A = \frac{l^2}{2} \)
Perimeter: \( P = 2l + h \)

Where \( l \) is the length of each leg.

4. Applications

Usage: Commonly used in construction, engineering, and design where right angles and equal proportions are needed.

5. Frequently Asked Questions (FAQ)

Q1: Why is the hypotenuse √2 times the leg?
A: This comes from the Pythagorean theorem: \( h = \sqrt{l^2 + l^2} = \sqrt{2l^2} = l\sqrt{2} \).

Q2: How do I find the legs if I know the hypotenuse?
A: Divide the hypotenuse by \( \sqrt{2} \): \( l = \frac{h}{\sqrt{2}} \).

Q3: Are all 45-45-90 triangles similar?
A: Yes, they are all similar by AA (Angle-Angle) similarity criterion.

Q4: What's special about the trigonometric values?
A: At 45°, sine and cosine have the same value, and tangent equals exactly 1.

Q5: Where is this triangle commonly found?
A: In squares (diagonal forms two 45-45-90 triangles), roof framing, and various engineering applications.