Finding the Smallest Angle in a Right Triangle with Given Side Lengths

In the realm of geometry, understanding the properties and dimensions of triangles is fundamental. One common problem is determining the smallest angle in a right triangle, given the lengths of its sides. For a right triangle with sides of 5 cm and an angle of 90 degrees between them, the calculation becomes straightforward using basic trigonometric principles. This article explores step-by-step how to find the smallest angle in such a triangle.

Understanding the Triangle

Consider a right triangle where one of the angles is 90 degrees, and the lengths of the two sides adjacent to this right angle are 5 cm and 8 cm. This triangle is a right-angled triangle with the sides 5 cm and 8 cm forming the right angle, and the hypotenuse being the longest side.

Calculating the Smallest Angle

The smallest angle in a triangle is always opposite the shortest side. In this case, the shortest side is 5 cm. To find the angle opposite to this side, we use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

Using the Tangent Function

The tangent of the smallest angle is given by the ratio of the opposite side (5 cm) to the adjacent side (8 cm). Therefore,

[ tan(X) frac{5}{8} ]

Using a calculator or trigonometric tables, we find that,

[ tan^{-1}left(frac{5}{8}right) approx 0.625 text{ or } 32° ]

Thus, the smallest angle of the triangle is approximately 32°.

Verification through Hypotenuse Calculation

For verification, we can calculate the hypotenuse using the Pythagorean theorem. The theorem states that in a right-angled triangle,

[ text{Hypotenuse}^2 5^2 8^2 ]

Calculating further,

[ text{Hypotenuse} sqrt{25 64} sqrt{89} approx 9.434 text{ cm} ]

Now, using the inverse sine function to find the smallest angle,

[ sin^{-1}left(frac{5}{9.434}right) approx 32.005° ]

This further confirms our previous calculation, showing that the smallest angle is indeed approximately 32°.

Additional Considerations

It's important to note that if the triangle is an isosceles right triangle, both base angles are 45°. However, in the scenario given, the sides are not equal, making it a right-angled triangle with different angles.

Conclusion

By understanding the basic principles of trigonometry and the properties of right-angled triangles, we can easily determine the smallest angle in a given right triangle. The smallest angle in a right triangle with sides of 5 cm and 8 cm is approximately 32°, as calculated using the tangent function and verified through the Pythagorean theorem.