Right Triangle Angles: Find Measures Easily

by Pedro Alvarez 44 views

Introduction

Hey guys! Let's dive into a classic geometry problem involving a right triangle. We've got a triangle ABC where side AC is 7 inches, BC is 24 inches, and AB is 25 inches. Our mission? To find the measures of all the angles in this triangle. This is a fantastic exercise in applying trigonometric principles and understanding the relationships between sides and angles in right triangles. So, buckle up, and let’s get started!

Verifying the Right Triangle

Before we jump into calculating angles, let's make sure we're dealing with a right triangle. How do we do that? The Pythagorean theorem is our best friend here! It 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. In mathematical terms: $a^2 + b^2 = c^2$.

In our case, AB is the longest side, so let's assume it's the hypotenuse. We need to check if $AC^2 + BC^2 = AB^2$. Plugging in the values, we get $7^2 + 24^2 = 49 + 576 = 625$. And what is $25^2$? It's also 625! So, we've confirmed that $7^2 + 24^2 = 25^2$, which means triangle ABC is indeed a right triangle. This is crucial because it allows us to use trigonometric ratios like sine, cosine, and tangent to find the angles. Knowing it’s a right triangle simplifies our calculations immensely. We know one angle is 90 degrees, and we can use trigonometric functions to find the other two. This verification step is essential in any geometry problem to ensure we're applying the correct principles and formulas.

Identifying the Right Angle

Now that we know we have a right triangle, we need to figure out which angle is the right angle (90 degrees). The right angle is always opposite the longest side, which is the hypotenuse. In our triangle, AB is the longest side (25 inches), so the angle opposite to AB is the right angle. That means angle C is the 90-degree angle. This identification is a fundamental step because it sets the stage for using trigonometric ratios correctly. The sides adjacent and opposite to the other angles are defined relative to this right angle. Without knowing which angle is the right angle, we can't accurately apply sine, cosine, and tangent to find the remaining angles. So, angle C is our cornerstone, the anchor for the rest of our calculations.

Using Trigonometric Ratios

Okay, with angle C confirmed as 90 degrees, let's find angles A and B. This is where trigonometry shines! We'll use the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) to relate the angles to the sides of the triangle. Remember the handy acronym SOH CAH TOA?

  • SOH: sin(angle) = Opposite / Hypotenuse
  • CAH: cos(angle) = Adjacent / Hypotenuse
  • TOA: tan(angle) = Opposite / Adjacent

Let's start with angle A. Relative to angle A:

  • BC is the opposite side (24 inches)
  • AC is the adjacent side (7 inches)
  • AB is the hypotenuse (25 inches)

We can use any of the trigonometric ratios, but let's use sine and tangent for variety. First, let's use the sine function. The sine of angle A is the ratio of the opposite side to the hypotenuse. So, $\sin(A) = \frac{BC}{AB} = \frac{24}{25}$. To find the measure of angle A, we need to take the inverse sine (also called arcsin) of $\frac{24}{25}$. Using a calculator, $A = \arcsin(\frac{24}{25}) \approx 73.74^{\circ}$.

Now, let's use the tangent function as a check and also to demonstrate another approach. The tangent of angle A is the ratio of the opposite side to the adjacent side. So, $\tan(A) = \frac{BC}{AC} = \frac{24}{7}$. To find the measure of angle A, we take the inverse tangent (arctan) of $\frac{24}{7}$. Using a calculator, $A = \arctan(\frac{24}{7}) \approx 73.74^{\circ}$. This confirms our earlier calculation using sine, which is always reassuring.

So, we've found that angle A is approximately 73.74 degrees. Now, let's move on to finding angle B.

Calculating Angle A

Let's get into the nitty-gritty of calculating angle A. As we discussed, we can use the trigonometric ratios to relate the sides and angles. For angle A, we identified that BC is the opposite side, AC is the adjacent side, and AB is the hypotenuse. This identification is crucial because the correct application of trigonometric functions depends on knowing which side is opposite, adjacent, and the hypotenuse relative to the angle we're interested in. Misidentifying these sides will lead to incorrect calculations, so always double-check your setup!

Using the sine function, we have: $\sin(A) = \frac{Opposite}{Hypotenuse} = \frac{BC}{AB} = \frac{24}{25}$. The sine function provides a ratio between the side opposite to the angle and the hypotenuse. It's a fundamental tool in trigonometry, allowing us to connect angles and side lengths in right triangles. This connection is what makes it possible to calculate unknown angles when we know the side lengths, and vice versa.

To find the angle A, we need to use the inverse sine function, also known as arcsin or $\sin^{-1}$. This function essentially