Trigonometric ratios for 45-45-90 triangles examples

📚 Quick Study Guide

• 📐 A 45-45-90 triangle is a special right triangle with two angles measuring 45 degrees and one angle measuring 90 degrees.
• 📏 The sides opposite the 45-degree angles (legs) are congruent (equal in length). Let's call the length of each leg 'a'.
• 🔗 The hypotenuse (the side opposite the 90-degree angle) is $\sqrt{2}$ times the length of a leg. So, if a leg has length 'a', the hypotenuse has length $a\sqrt{2}$.
• ➗ Sine (sin) of 45 degrees: $\sin(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.
• ➗ Cosine (cos) of 45 degrees: $\cos(45^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.
• ➗ Tangent (tan) of 45 degrees: $\tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{a} = 1$.

Practice Quiz

1. What is the sine of 45 degrees in a 45-45-90 triangle?
   1. $\frac{1}{2}$
   2. 1
   3. $\frac{\sqrt{2}}{2}$
   4. $\sqrt{2}$
2. In a 45-45-90 triangle, if one leg has a length of 5, what is the length of the hypotenuse?
   1. 5
   2. $5\sqrt{2}$
   3. 10
   4. $5\sqrt{3}$
3. What is the cosine of 45 degrees in a 45-45-90 triangle?
   1. 1
   2. $\frac{1}{2}$
   3. $\frac{\sqrt{3}}{2}$
   4. $\frac{\sqrt{2}}{2}$
4. What is the tangent of 45 degrees in a 45-45-90 triangle?
   1. $\frac{\sqrt{2}}{2}$
   2. $\sqrt{2}$
   3. 1
   4. 0
5. If the hypotenuse of a 45-45-90 triangle is $7\sqrt{2}$, what is the length of each leg?
   1. $\frac{7\sqrt{2}}{2}$
   2. 7
   3. 14
   4. $\frac{7}{2}$
6. In a 45-45-90 triangle, if the length of one leg is x, what is the length of the other leg?
   1. $x\sqrt{2}$
   2. $\frac{x}{\sqrt{2}}$
   3. x
   4. 2x
7. Which of the following statements is true for a 45-45-90 triangle?
   1. All sides are equal.
   2. The two legs are equal and the hypotenuse is equal to the length of one leg.
   3. The two legs are equal and the hypotenuse is $\sqrt{2}$ times the length of one leg.
   4. All angles are equal.