Understanding the Length of the Longer Leg in a 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the sides have a specific ratio. If the hypotenuse is known, the lengths of the other two sides can be easily calculated. This article will explore how to determine the length of the longer leg in such a triangle. We will discuss various methods and provide a detailed explanation with examples.

The Importance of Knowing Specific Sine Values for 45°-30°-60° Triangles

Understanding the sine values for specific angles (such as 30°, 45°, and 60°) is crucial for solving problems involving 30°-60°-90° triangles. For 30°-60°-90° triangles, the sine of 60° is commonly required. It is essential to look up these values and remember them for quick and accurate problem-solving. By knowing that sin(60°) √3/2, you can solve for the longer leg of the triangle.

Calculating the Longer Leg Using the Sine Rule

A common method to find the length of the longer leg (opposite the 60° angle) is to use the sine rule. Given a hypotenuse of 12, the sine rule states:

sin(60°) x / 12

Solving for x:

x 12 * sin(60°) 12 * (√3 / 2) 6√3

Thus, the longer leg is 6√3 units. This value can be approximated to 10.392 units.

Drawing the Triangle for Clarity

When faced with problems involving 30°-60°-90° triangles, it is always advisable to draw the triangle and label it with all necessary details. In this case:

Draw an angle of 30° and an angle of 60°, making sure the 30° angle is smaller than the 60° angle. Label the hypotenuse as 12. Near the 60° angle, draw the longer leg and label it as x. Near the 30° angle, draw the shorter leg and label it as y.

By drawing and labeling the triangle correctly, you can visualize and solve the problem more effectively.

Using Trigonometric Ratios for 30°-60°-90° Triangles

The sides of a 30°-60°-90° triangle have a specific ratio. The shorter leg (opposite the 30° angle) is half the length of the hypotenuse, and the longer leg (opposite the 60° angle) is the shorter leg times the square root of 3.

Given a hypotenuse of 12:

The shorter leg (opposite 30°) 12 / 2 6 units. The longer leg (opposite 60°) 6 * √3 6√3 units.

Therefore, the longer leg is 6√3 units, which is approximately 10.392 units.

Verifying the Solution Using the Sine Law

To ensure the correctness of the solution, we can use the sine law. Given angles A (30°), B (60°), and C (90°), and the hypotenuse AC of 12:

sin(60°) AB / 12

Note that sin(60°) √3 / 2, so:

(√3 / 2) AB / 12

Solving for AB:

AB 12 * (√3 / 2) 6√3

Thus, the longer leg AB is 6√3 units, or approximately 10.392 units.

In conclusion, understanding and applying these principles of 30°-60°-90° triangles can greatly aid in problem-solving. Whether using the sine rule or the specific trigonometric ratios, the key to success lies in clear understanding and proper application.