Find The Tangent Of The Larger Acute Angle In A Right Triangle With Side Lengths 3, 4, And 5. Write Your Answer As A Fraction In Simplest Form.The Tangent Of The Larger Acute Angle Is:

by ADMIN 185 views

Introduction

In a right triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Given a right triangle with side lengths 3, 4, and 5, we need to find the tangent of the larger acute angle. To do this, we will first identify the sides of the triangle and then use the definition of tangent to calculate the desired value.

Understanding the Triangle

The given right triangle has side lengths 3, 4, and 5. Since the sum of the squares of the two shorter sides is equal to the square of the hypotenuse (by the Pythagorean theorem), we can confirm that this is indeed a right triangle. The side lengths 3, 4, and 5 satisfy the Pythagorean theorem, as 3^2 + 4^2 = 5^2.

Identifying the Sides

Let's identify the sides of the triangle. The side with length 5 is the hypotenuse, as it is opposite the right angle. The side with length 3 is one of the legs, and the side with length 4 is the other leg.

Calculating the Tangent of the Larger Acute Angle

To calculate the tangent of the larger acute angle, we need to find the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Since the larger acute angle is opposite the side with length 4, we will use the side with length 3 as the adjacent side.

The tangent of the larger acute angle is given by:

tan(θ) = opposite side / adjacent side = 4 / 3

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4 and 3 is 1, so the fraction is already in its simplest form.

Conclusion

In conclusion, the tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5 is 4/3.

Additional Information

It's worth noting that the tangent function is periodic, meaning that it repeats its values at regular intervals. In this case, the tangent function has a period of π (180 degrees). This means that the tangent of the larger acute angle is equal to the tangent of the angle that is π radians (180 degrees) larger or smaller.

Real-World Applications

The tangent function has many real-world applications, including:

  • Trigonometry: The tangent function is used to solve triangles and find unknown side lengths and angles.
  • Physics: The tangent function is used to describe the motion of objects and the forces acting on them.
  • Engineering: The tangent function is used to design and analyze mechanical systems, such as gears and linkages.

Final Thoughts

In this article, we have seen how to find the tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5. We have also discussed the importance of the tangent function in real-world applications and its periodic nature. By understanding the tangent function and its properties, we can solve a wide range of in mathematics and physics.

References

  • "Trigonometry" by Michael Corral
  • "Physics for Scientists and Engineers" by Paul A. Tipler
  • "Engineering Mechanics" by Russell C. Hibbeler

Related Topics

  • Right Triangles: A right triangle is a triangle with one right angle (90 degrees).
  • Tangent Function: The tangent function is a trigonometric function that describes the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
  • Pythagorean Theorem: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle.

Keywords

  • Tangent: The tangent function is a trigonometric function that describes the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
  • Right Triangle: A right triangle is a triangle with one right angle (90 degrees).
  • Pythagorean Theorem: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle.

Introduction

In our previous article, we discussed how to find the tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is the tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5?

A: The tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5 is 4/3.

Q: How do I find the tangent of the larger acute angle in a right triangle?

A: To find the tangent of the larger acute angle in a right triangle, you need to identify the sides of the triangle and then use the definition of tangent to calculate the desired value. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: What is the difference between the tangent and the sine of an angle?

A: The tangent and the sine of an angle are both trigonometric functions, but they are defined differently. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: Can I use the tangent function to solve triangles with non-right angles?

A: Yes, you can use the tangent function to solve triangles with non-right angles. However, you need to use the law of sines or the law of cosines to find the lengths of the sides of the triangle.

Q: What is the relationship between the tangent and the cotangent functions?

A: The tangent and the cotangent functions are reciprocal functions. The cotangent of an angle is defined as the reciprocal of the tangent of the angle.

Q: Can I use the tangent function to find the length of the hypotenuse of a right triangle?

A: Yes, you can use the tangent function to find the length of the hypotenuse of a right triangle. However, you need to use the Pythagorean theorem to find the length of the hypotenuse.

Q: What is the significance of the tangent function in real-world applications?

A: The tangent function has many real-world applications, including trigonometry, physics, and engineering. It is used to solve triangles, find unknown side lengths and angles, and describe the motion of objects and the forces acting on them.

Q: Can I use the tangent function to solve problems in other areas of mathematics, such as algebra and geometry?

A: Yes, you can use the tangent function to solve problems in other areas of mathematics, such as algebra and geometry. However, you need to use the tangent function in conjunction with other mathematical concepts and techniques.

Q: What are some common mistakes to avoid when using the tangent function?

A: Some common mistakes to avoid when using the tangent function include:

  • Using the tangent function to solve triangles with non-right angles without using the law of sines or law of cosines.
  • Using the tangent function to find the length of the hypotenuse of a right triangle without using the Pythagorean theorem.
  • Using the tangent function to solve problems in other areas of mathematics without using the tangent function in conjunction with other mathematical concepts and techniques.

Conclusion

In conclusion, the tangent function is a powerful tool for solving triangles and finding unknown side lengths and angles. However, it is essential to use the tangent function correctly and avoid common mistakes. By understanding the tangent function and its properties, you can solve a wide range of problems in mathematics and physics.

Additional Resources

  • "Trigonometry" by Michael Corral
  • "Physics for Scientists and Engineers" by Paul A. Tipler
  • "Engineering Mechanics" by Russell C. Hibbeler

Related Topics

  • Right Triangles: A right triangle is a triangle with one right angle (90 degrees).
  • Tangent Function: The tangent function is a trigonometric function that describes the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
  • Pythagorean Theorem: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle.

Keywords

  • Tangent: The tangent function is a trigonometric function that describes the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
  • Right Triangle: A right triangle is a triangle with one right angle (90 degrees).
  • Pythagorean Theorem: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle.