Find Angles Of Right Triangle ABC: A Step-by-Step Guide

Hey there, math enthusiasts! Ever find yourself staring at a right triangle, scratching your head wondering how to figure out those tricky angles? Well, you're in the right place! Let's break down a classic problem: a right triangle ABC with sides AC = 7 inches, BC = 24 inches, and AB = 25 inches. Our mission? To uncover the measures of angles A, B, and C.

Understanding the Basics of Right Triangles

Before we dive into the calculations, let’s refresh some fundamental concepts. A right triangle is a triangle containing one angle that measures exactly 90 degrees. This angle is often denoted by a small square at the vertex. The side opposite the right angle is called the hypotenuse, which is also the longest side of the triangle. The other two sides are known as the legs.

In our triangle ABC, side AB (25 inches) is the hypotenuse because it’s the longest side and likely opposite the right angle (we will confirm this shortly!). Sides AC (7 inches) and BC (24 inches) are the legs. The Pythagorean theorem is a crucial principle for right triangles, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, this is expressed as: $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.

Confirming the Right Angle

To ensure triangle ABC is indeed a right triangle, let’s apply the Pythagorean theorem using the given side lengths. We'll check if $AC^2 + BC^2 = AB^2$:

• $AC^2 = 7^2 = 49$
• $BC^2 = 24^2 = 576$
• $AB^2 = 25^2 = 625$

Adding the squares of the legs:

$49 + 576 = 625$

Since $AC^2 + BC^2 = AB^2$, we can confidently confirm that triangle ABC is a right triangle. This means that angle C, opposite the hypotenuse AB, is the right angle. Therefore, $m \angle C = 90^{\circ}$. This is a critical first step in solving the problem, as it allows us to use trigonometric ratios to find the other angles.

Trigonometric Ratios: SOH-CAH-TOA

Now that we know one angle, we can determine the other two using trigonometric ratios. Remember the acronym SOH-CAH-TOA? It’s your best friend when dealing with right triangles:

• SOH: Sine (sin) = Opposite / Hypotenuse
• CAH: Cosine (cos) = Adjacent / Hypotenuse
• TOA: Tangent (tan) = Opposite / Adjacent

These ratios relate the angles of a right triangle to the ratios of its sides. Let's use these to find the measures of angles A and B.

Finding Angle A

To find the measure of angle A ($m \angle A$), we can use any of the trigonometric ratios. Let's use the sine function for this example. From the perspective of angle A, the opposite side is BC (24 inches), and the hypotenuse is AB (25 inches). Therefore:

$sin(A) = \frac{Opposite}{Hypotenuse} = \frac{BC}{AB} = \frac{24}{25}$

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

$A = sin^{-1}(\frac{24}{25})$

Using a calculator, we find:

$A \approx 73.74^{\circ}$

Finding Angle B

Now, let's find the measure of angle B ($m \angle B$). We can use the cosine function this time. From the perspective of angle B, the adjacent side is BC (24 inches), and the hypotenuse is AB (25 inches). Therefore:

$cos(B) = \frac{Adjacent}{Hypotenuse} = \frac{BC}{AB} = \frac{24}{25}$

To find angle B, we use the inverse cosine function, also known as arccos or $cos^{-1}$:

$B = cos^{-1}(\frac{24}{25})$

Using a calculator, we get:

$B \approx 16.26^{\circ}$

Alternative Method: Using the Triangle Angle Sum Theorem

Alternatively, we could have found angle B using the Triangle Angle Sum Theorem, which states that the sum of the angles in any triangle is always 180 degrees. Since we already know angle A (approximately 73.74 degrees) and angle C (90 degrees), we can find angle B as follows:

$m \angle A + m \angle B + m \angle C = 180^{\circ}$

$73.74^{\circ} + m \angle B + 90^{\circ} = 180^{\circ}$

$m \angle B = 180^{\circ} - 73.74^{\circ} - 90^{\circ}$

$m \angle B \approx 16.26^{\circ}$

This method serves as a great check to ensure our trigonometric calculations are accurate!

Summarizing the Angle Measures

Let's compile our findings. We've determined the measures of the angles in triangle ABC:

• $m \angle A \approx 73.74^{\circ}$
• $m \angle B \approx 16.26^{\circ}$
• $m \angle C = 90^{\circ}$

So, there you have it! We've successfully navigated the trigonometric landscape of triangle ABC to find all its angles. Remember, the key is to understand the definitions of trigonometric ratios (SOH-CAH-TOA) and how to apply them using the inverse trigonometric functions.

Reflecting on Common Mistakes

When working with trigonometric ratios, one common mistake is using the wrong sides relative to the angle. Always double-check which side is opposite, adjacent, and the hypotenuse with respect to the angle you're analyzing. Another mistake is forgetting to use the inverse trigonometric functions when solving for angles. Make sure you're using $sin^{-1}$, $cos^{-1}$, or $tan^{-1}$ when needed.

Conclusion: Mastering Right Triangle Trigonometry

By breaking down the problem step-by-step, we've shown how to find the angles of a right triangle using the Pythagorean theorem and trigonometric ratios. This knowledge is not only essential for mathematics but also for various real-world applications in fields like engineering, physics, and navigation. Keep practicing, and you'll become a right triangle trigonometry pro in no time!

For more in-depth information and practice problems on right triangle trigonometry, check out trusted resources like Khan Academy's Trigonometry Section. Happy calculating!