Understanding the 30-60-90 Right-Angled Triangle

In a right-angled triangle with angles of 30°, 60°, and 90°, the sides have a specific ratio based on the angles. The side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is (frac{sqrt{3}}{2}) times the length of the hypotenuse.

Calculating the Sides with Given Hypotenuse

Given that the hypotenuse of the triangle is 2 meters (m), we can determine the lengths of the other two sides as follows:

Side Opposite the 30° Angle:

Length (frac{1}{2} times) hypotenuse (frac{1}{2} times 2) m 1 m

Side Opposite the 60° Angle:

Length (frac{sqrt{3}}{2} times) hypotenuse (frac{sqrt{3}}{2} times 2) m (sqrt{3}) m, which is approximately 1.73 m

Alternative Methods for Finding the Sides

One can also use the sine function to find the lengths of these sides:

Given the hypotenuse h 2 m,

(sin 30° frac{p}{h} Rightarrow sin 30° frac{p}{2}) m (frac{1}{2}) m, thus:

(p 1) m

(cos 30° frac{b}{h} Rightarrow cos 30° frac{b}{2}) m (frac{sqrt{3}}{2}) m, thus:

(b sqrt{3}) m

The sides of a 30-60-90 triangle follow a specific ratio: the shortest side is (a), the middle side is (asqrt{3}), and the hypotenuse is (2a). Given the hypotenuse is 2 m, we find the shortest side (a)

Trigonometric Identities and Pythagoras' Theorem

Using trigonometric identities, we can confirm these results:

(sin 30° frac{1}{2} Rightarrow frac{1}{2} frac{p}{2}) m, thus:

(p 1) m, which is the side opposite the 30° angle.

Similarly,

(cos 30° frac{sqrt{3}}{2} Rightarrow frac{sqrt{3}}{2} frac{b}{2}) m, thus:

(b sqrt{3}) m, which is the side opposite the 60° angle.

Using the Pythagorean Theorem

The Pythagorean theorem states that (a^2 b^2 c^2), where (c) is the hypotenuse, and (a) and (b) are the other two sides. Substituting the known values:

(a^2 b^2 (2)^2 Rightarrow 1^2 b^2 4 Rightarrow b^2 3)

(b sqrt{3}) m, which is the side opposite the 60° angle.

These methods confirm that the sides of the triangle are:

Side opposite 30°: 1 m Side opposite 60°: (sqrt{3}) m, approximately 1.73 m Hypotenuse: 2 m

Conclusion

Understanding the properties of a 30-60-90 right-angled triangle is crucial for solving problems in trigonometry and geometry. Whether you use the specific ratio of the sides or trigonometric identities and the Pythagorean theorem, the results are consistent.

For more detailed trigonometric concepts, refer to this resource.