Recalling Trigonometric Concepts

Oct 13, 2025 by miodeya.com 29 views

Introduction

In this article, we will explore a problem involving a right triangle with a known shortest side and a specific angle. We will use trigonometric concepts to find the length of the hypotenuse of the triangle. This problem is a great example of how trigonometry can be applied to real-world scenarios.

Understanding the Problem

The problem states that the shortest side of a right triangle measures $$3 \sqrt{3}$$ inches. One angle of the triangle measures $$60^{\circ}$$. We are asked to find the length, in inches, of the hypotenuse of the triangle.

Recalling Trigonometric Concepts

To solve this problem, we need to recall some basic trigonometric concepts. In a right triangle, the sine, cosine, and tangent of an angle are defined as follows:

Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side.

Applying Trigonometric Concepts to the Problem

Since we are given the length of the shortest side and one angle, we can use the sine function to find the length of the hypotenuse. The sine of $$60^{\circ}$$ is $$\frac{\sqrt{3}}{2}$$. We can set up the following equation:

\sin(60^{\circ}) = \frac{\text{opposite side}}{\text{hypotenuse}}

We know that the opposite side is $$3 \sqrt{3}$$ inches, so we can substitute this value into the equation:

\frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{\text{hypotenuse}}

Solving for the Hypotenuse

To solve for the hypotenuse, we can cross-multiply and simplify the equation:

\text{hypotenuse} = \frac{2 \cdot 3 \sqrt{3}}{\sqrt{3}}

\text{hypotenuse} = 6

Therefore, the length of the hypotenuse of the triangle is 6 inches.

Conclusion

In this article, we used trigonometric concepts to solve a problem involving a right triangle with a known shortest side and a specific angle. We applied the sine function to find the length of the hypotenuse, and we arrived at the correct answer of 6 inches.

Real-World Applications

This problem has real-world applications in various fields, such as engineering, physics, and architecture. For example, in building design, architects need to calculate the length of hypotenuses to ensure that structures are stable and secure.

Tips and Tricks

When solving problems involving right triangles, it's essential to recall the definitions of sine, cosine, and tangent. Additionally, make sure to use the correct trigonometric function to solve the problem.

Common Mistakes

One common mistake when solving problems involving right triangles is to confuse the sine, cosine, and tangent functions. Make sure to use the correct function to solve the problem.

Practice Problems

Here are some practice problems to help you reinforce your understanding of trigonometry:

Find the length of the hypotenuse of a right triangle with a shortest side of 4 inches and an angle of $$30^{\circ}$$.
Find the length of the hypotenuse of a right triangle with a shortest side of 5 inches and an angle of $$45^{\circ}$$.

Conclusion

Q: What is the definition of a right triangle?

A: A right triangle is a triangle with one angle that measures 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Q: What is the sine, cosine, and tangent of an angle in a right triangle?

A: The sine, cosine, and tangent of an angle in a right triangle are defined as follows:

Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side.

Q: How do I use the sine, cosine, and tangent functions to solve right triangle problems?

A: To use the sine, cosine, and tangent functions to solve right triangle problems, you need to identify the given information and determine which function to use. For example, if you are given the length of the hypotenuse and the angle, you can use the sine function to find the length of the opposite side.

Q: What is the difference between the sine, cosine, and tangent functions?

A: The sine, cosine, and tangent functions are all ratios of the sides of a right triangle, but they are defined differently. The sine function is the ratio of the length of the opposite side to the length of the hypotenuse, the cosine function is the ratio of the length of the adjacent side to the length of the hypotenuse, and the tangent function is the ratio of the length of the opposite side to the length of the adjacent side.

Q: How do I solve a right triangle problem involving a 30-60-90 triangle?

A: To solve a right triangle problem involving a 30-60-90 triangle, you need to recall the special properties of this type of triangle. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is $$\frac{\sqrt{3}}{2}$$ times the length of the hypotenuse.

Q: What is the formula for finding the length of the hypotenuse of a right triangle?

A: The formula for finding the length of the hypotenuse of a right triangle is:

\text{hypotenuse} = \sqrt{\text{leg}_1^2 + \text{leg}_2^2}

Q: How do I use the Pythagorean theorem to solve right triangle problems?

A: To use the Pythagorean theorem to solve right triangle problems, you need to identify the given information and determine which sides of the triangle are the legs and the hypotenuse. Then, you can plug the values into the formula and solve for the unknown side.

Q: What are some common mistakes to avoid when solving right triangle problems?

A: Some common mistakes to avoid when solving right triangle problems include:

Confusing the sine, cosine, and tangent functions
Not identifying the given information and determining which function to use
Not using the correct formula or formula variation
Not checking the units of the answer

Q: How can I practice solving right triangle problems?

A: You can practice solving right triangle problems by working through example problems, using online resources, and taking practice quizzes. It's also a good idea to review the concepts and formulas regularly to reinforce your understanding.

Conclusion

In conclusion, solving right triangle problems requires a strong understanding of trigonometry and the ability to apply the sine, cosine, and tangent functions. By recalling the definitions of these functions and using the correct formulas, you can solve a wide range of right triangle problems. Remember to practice regularly and avoid common mistakes to become proficient in solving these types of problems.