Solving 30°-60°-90° Right Triangle Given the Hypotenuse

Triangles, especially right triangles, are of significant importance in geometry and trigonometry. Among the various types of right triangles, 30°-60°-90° triangles have unique properties that can simplify solving problems involving them. This article will guide you through how to solve a 30°-60°-90° right triangle when given only the length of the hypotenuse, along with some fundamental trigonometric principles.

Introduction to 30°-60°-90° Triangles

A 30°-60°-90° triangle is a special right triangle where the angles are 30°, 60°, and 90°. The sides of this triangle are in a specific ratio: 1: √3: 2. This means that if the hypotenuse is (c), the side opposite the 30° angle is (c/2) and the side opposite the 60° angle is (csqrt{3}/2).

Solving the Triangle Given the Hypotenuse

Given only the length of the hypotenuse ((c)), we can find the lengths of the other two sides using the consistent ratio of 1:√3:2.

Step-by-Step Solution

1. Identify Side (a):
(a) is the side opposite the 30° angle, which is half the length of the hypotenuse:

(a frac{c}{2})

2. Identify Side (b):
(b) is the side opposite the 60° angle, which is (asqrt{3}) or (frac{csqrt{3}}{2}):

(b frac{csqrt{3}}{2})

Example

Suppose the hypotenuse (c) is 10 units:

1. Calculate (a):
(a frac{10}{2} 5)

2. Calculate (b):
(b frac{10sqrt{3}}{2} 5sqrt{3} approx 8.66)

Therefore, the sides of the triangle are approximately 5 units opposite the 30° angle and (5sqrt{3}) units opposite the 60° angle.

Trigonometric Functions of 30° and 60°

Additionally, the trigonometric functions of 30° and 60° can be determined using this triangle.

Sine and Cosine Functions

Sine of 30°: (sin 30° frac{1}{2} frac{text{opposite}}{text{hypotenuse}} frac{a}{c} frac{c/2}{c} frac{1}{2})

Cosine of 30°: (cos 30° frac{sqrt{3}}{2} frac{text{adjacent}}{text{hypotenuse}} frac{sqrt{3}c/2}{c} frac{sqrt{3}}{2})

Tangent Function

Tangent of 30°: (tan 30° frac{text{opposite}}{text{adjacent}} frac{frac{c}{2}}{frac{csqrt{3}}{2}} frac{1}{sqrt{3}} frac{sqrt{3}}{3})

Tangent of 60°: (tan 60° frac{text{opposite}}{text{adjacent}} frac{frac{csqrt{3}}{2}}{frac{c}{2}} sqrt{3})

Conclusion

Solving 30°-60°-90° triangles given the hypotenuse is a fundamental skill in geometry and trigonometry. By understanding the consistent 1:√3:2 ratio and the basic trigonometric identities for 30° and 60°, you can easily solve such problems and apply these principles to more complex mathematical and real-world scenarios.