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:
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. This involves using trigonometric ratios and applying them to the given triangle.
Understanding the Triangle
The given right triangle has side lengths 3, 4, and 5. We can use the Pythagorean theorem to verify that this is indeed a right triangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Applying the Pythagorean Theorem
Let's apply the Pythagorean theorem to the given triangle:
c² = a² + b²
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
c² = 3² + 4² c² = 9 + 16 c² = 25 c = √25 c = 5
This confirms that the given triangle is indeed a right triangle with side lengths 3, 4, and 5.
Identifying the Larger Acute Angle
In a right triangle, the larger acute angle is the angle that is opposite the longer of the two legs (the sides that are not the hypotenuse). In this case, the longer leg is 4, so the larger acute angle is the angle opposite the side of length 4.
Finding the Tangent of the Larger Acute Angle
To find the tangent of the larger acute angle, we need to use the definition of the tangent function:
tan(θ) = opposite side / adjacent side
where θ is the angle, and the opposite side and adjacent side are the sides of the triangle that are opposite and adjacent to the angle, respectively.
In this case, the opposite side is 4, and the adjacent side is 3. Therefore, the tangent of the larger acute angle is:
tan(θ) = 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 we cannot simplify the fraction further.
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
- The tangent function is a trigonometric function that is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
- The tangent function is used to find the ratio of the opposite side to the adjacent side in a right triangle.
- The tangent function is a fundamental concept in trigonometry and is used to solve problems involving right triangles.
Example Problems
- Find the tangent of the larger acute angle in a right triangle with side lengths 5, 12, and 13.
- Find the tangent of the smaller acute angle a right triangle with side lengths 3, 4, and 5.
Solutions to Example Problems
- To find the tangent of the larger acute angle in a right triangle with side lengths 5, 12, and 13, we can use the definition of the tangent function: tan(θ) = opposite side / adjacent side tan(θ) = 12 / 5
- To find the tangent of the smaller acute angle in a right triangle with side lengths 3, 4, and 5, we can use the definition of the tangent function: tan(θ) = opposite side / adjacent side tan(θ) = 3 / 4
Final Thoughts
In conclusion, the tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5 is 4/3. This involves using trigonometric ratios and applying them to the given triangle. The tangent function is a fundamental concept in trigonometry and is used to solve problems involving right triangles.
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. We used the definition of the tangent function and applied it to the given triangle to find the tangent of the larger acute angle. In this article, we will answer some frequently asked questions related to this topic.
Q&A
Q: What is the tangent of the smaller acute angle in a right triangle with side lengths 3, 4, and 5?
A: To find the tangent of the smaller acute angle, we can use the definition of the tangent function: tan(θ) = opposite side / adjacent side tan(θ) = 3 / 4
Q: How do I find the tangent of an angle in a right triangle?
A: To find the tangent of an angle in a right triangle, you can use the definition of the tangent function: tan(θ) = opposite side / adjacent side where θ is the angle, and the opposite side and adjacent side are the sides of the triangle that are opposite and adjacent to the angle, respectively.
Q: What is the difference between the tangent and the sine of an angle in a right triangle?
A: The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, while the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
Q: Can I use the tangent function to find the length of a side in a right triangle?
A: Yes, you can use the tangent function to find the length of a side in a right triangle. If you know the length of the opposite side and the adjacent side, you can use the tangent function to find the length of the hypotenuse.
Q: How do I find the tangent of an angle in a right triangle when the lengths of the sides are not given?
A: If the lengths of the sides are not given, you can use the Pythagorean theorem to find the length of the hypotenuse, and then use the definition of the tangent function to find the tangent of the angle.
Q: Can I use the tangent function to find the angle in a right triangle?
A: Yes, you can use the tangent function to find the angle in a right triangle. If you know the length of the opposite side and the adjacent side, you can use the tangent function to find the angle.
Example Problems
- Find the tangent of the larger acute angle in a right triangle with side lengths 5, 12, and 13.
- Find the tangent of the smaller acute angle in a right triangle with side lengths 3, 4, and 5.
- Find the length of the hypotenuse in a right triangle with side lengths 3, 4, and 5.
Solutions to Example Problems
- To find the tangent of the larger acute angle in a right triangle with side lengths 5, 12, and 13, we can use the definition of the tangent function: tan(θ) = opposite side / adjacent side tan(θ) =12 / 5
- To find the tangent of the smaller acute angle in a right triangle with side lengths 3, 4, and 5, we can use the definition of the tangent function: tan(θ) = opposite side / adjacent side tan(θ) = 3 / 4
- To find the length of the hypotenuse in a right triangle with side lengths 3, 4, and 5, we can use the Pythagorean theorem: c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25 c = √25 c = 5
Final Thoughts
In conclusion, the tangent of the larger acute angle in a right triangle with side lengths 3, 4, and 5 is 4/3. We also answered some frequently asked questions related to this topic, including how to find the tangent of an angle in a right triangle, the difference between the tangent and the sine of an angle in a right triangle, and how to use the tangent function to find the length of a side in a right triangle.