A Right Triangle Has Side Lengths { AC = 7$}$ Inches, { BC = 24$}$ Inches, And { AB = 25$}$ Inches. What Are The Measures Of The Angles In Triangle ABC?A. { M \angle A = 46.2^{\circ}$} , \[ , \[ , \[ M \angle B =
Introduction
In this article, we will delve into the world of trigonometry and explore how to find the measures of angles in a right triangle. A right triangle is a triangle with one angle that measures 90 degrees. The side lengths of the triangle are given as {AC = 7$}$ inches, {BC = 24$}$ inches, and {AB = 25$}$ inches. We will use the Pythagorean theorem and trigonometric ratios to find the measures of the angles in triangle ABC.
Understanding the Pythagorean Theorem
The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled 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. Mathematically, this can be expressed as:
a^2 + b^2 = c^2
where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.
Applying the Pythagorean Theorem to Triangle ABC
In triangle ABC, we are given the side lengths as {AC = 7$}$ inches, {BC = 24$}$ inches, and {AB = 25$}$ inches. We can use the Pythagorean theorem to verify that this is indeed a right triangle.
a^2 + b^2 = c^2 7^2 + 24^2 = 25^2 49 + 576 = 625 625 = 625
As we can see, the equation holds true, which confirms that triangle ABC is a right triangle.
Finding the Measures of Angles in Triangle ABC
Now that we have verified that triangle ABC is a right triangle, we can use trigonometric ratios to find the measures of the angles in the triangle. The trigonometric ratios that we will use are:
- Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
- Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Finding the Measure of Angle A
To find the measure of angle A, we can use the sine ratio:
sin(A) = opposite side / hypotenuse sin(A) = 7 / 25 sin(A) = 0.28
Using a calculator, we can find the inverse sine of 0.28 to get the measure of angle A:
m ∠A = arcsin(0.28) m ∠A = 16.26°
Finding the Measure of Angle B
To find the measure of angle B, we can use the sine ratio:
sin(B) = opposite side / hypotenuse sin(B) = 24 / 25 sin(B) = 0.96
Using a calculator, we can find the inverse sine of 0.96 to get the measure of angle B:
m ∠B = arcsin(0.96) m ∠B = 83.68°
Finding the Measure of Angle C
Since triangle ABC is a right triangle, we know that the measure of angle C is 90 degrees.
m ∠C = 90°
Conclusion
In this article, we have explored how to find the measures of angles in a right triangle using the Pythagorean theorem and trigonometric ratios. We have used the given side lengths of triangle ABC to find the measures of the angles in the triangle. The measures of the angles in triangle ABC are:
- m ∠A = 16.26°
- m ∠B = 83.68°
- m ∠C = 90°
Introduction
In our previous article, we explored how to find the measures of angles in a right triangle using the Pythagorean theorem and trigonometric ratios. In this article, we will answer some of the most frequently asked questions related to finding the measures of angles in a right triangle.
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled 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. Mathematically, this can be expressed as:
a^2 + b^2 = c^2
where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.
Q: How do I use the Pythagorean theorem to find the measures of angles in a right triangle?
A: To use the Pythagorean theorem to find the measures of angles in a right triangle, you need to first verify that the triangle is a right triangle by checking if the equation a^2 + b^2 = c^2 holds true. Once you have verified that the triangle is a right triangle, you can use trigonometric ratios to find the measures of the angles in the triangle.
Q: What are the trigonometric ratios that I can use to find the measures of angles in a right triangle?
A: The trigonometric ratios that you can use to find the measures of angles in a right triangle are:
- Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
- Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Q: How do I use the sine ratio to find the measure of an angle in a right triangle?
A: To use the sine ratio to find the measure of an angle in a right triangle, you need to first identify the side opposite the angle and the hypotenuse. Then, you can use the sine ratio to find the measure of the angle:
sin(A) = opposite side / hypotenuse
Using a calculator, you can find the inverse sine of the ratio to get the measure of the angle.
Q: How do I use the cosine ratio to find the measure of an angle in a right triangle?
A: To use the cosine ratio to find the measure of an angle in a right triangle, you need to first identify the side adjacent to the angle and the hypotenuse. Then, you can use the cosine ratio to find the measure of the angle:
cos(A) = adjacent side / hypotenuse
Using a calculator, you can find the inverse cosine of the ratio to get the measure of the angle.
Q: How do I use the tangent ratio to find the measure of an angle in a right triangle?
A: To use the tangent ratio to find the measure of an angle in a right triangle, you need to first identify the side opposite the angle and the side adjacent to the angle. Then, you can use the tangent ratio to find the measure of the angle:
tan(A) = opposite side / adjacent side
Using a calculator, you can find the inverse tangent of the ratio to get the measure of the angle.
Q: What are some common mistakes to avoid when finding the measures of angles in a right triangle?
A: Some common mistakes to avoid when finding the measures of angles in a right triangle include:
- Not verifying that the triangle is a right triangle: Make sure to verify that the triangle is a right triangle by checking if the equation a^2 + b^2 = c^2 holds true.
- Not using the correct trigonometric ratio: Make sure to use the correct trigonometric ratio to find the measure of the angle.
- Not using a calculator to find the inverse trigonometric function: Make sure to use a calculator to find the inverse trigonometric function to get the measure of the angle.
Conclusion
In this article, we have answered some of the most frequently asked questions related to finding the measures of angles in a right triangle. We hope that this article has provided a comprehensive guide to finding the measures of angles in a right triangle and has helped you to avoid common mistakes.