Problem 18 - Algebra 2

Question

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(b=5, c=10\)

Step-by-Step Solution

Answer

The remaining side a of the triangle is approximately 8.7 units, angle A is 30 degrees, and angle B is 60 degrees.

Step 1: Find side a

First, find the remaining side using the Pythagorean theorem, \(a^2 + b^2 = c^2\). Given \(b = 5\) and \(c = 10\), we get \(a^2 + 5^2 = 10^2\), or \(a^2 = 75\). Taking the square root of both sides, \(a = \sqrt{75} \approx 8.7\), rounding to the nearest tenth.

Step 2: Find angle A

Use the sine function to find angle A. The sine of an angle in a right triangle is defined as the length of the side opposite the angle divided by the length of the hypotenuse. Therefore, \(\sin(A) = \frac{b}{c} = \frac{5}{10} = 0.5\). Use the inverse sine function to find the measure of angle A. \(\sin^{-1}(0.5) = 30^\circ\)

Step 3: Find angle B

The third angle of a triangle is found by subtracting the other two angles from 180 degrees. So, \(\angle B = 180^\circ - 90^\circ - 30^\circ = 60^\circ\)

Step 4: Summary

All the sides and angles of the triangle are now known. Side a is approximately 8.7 units, \(\angle A\) is \(30^\circ\), and \(\angle B\) is \(60^\circ\).

Key Concepts

Pythagorean Theorem

A right triangle has one angle at 90 degrees, which makes it unique in many ways. One of the key properties of right triangles is that they adhere to the Pythagorean Theorem. This theorem is essential for finding the lengths of sides in right triangles. 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. The formula is written as:\[ a^2 + b^2 = c^2 \]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. In our example with \(b = 5\) and \(c = 10\), you can plug these values into the equation to find \(a\).

• Calculate \(b^2 = 25\).
• Substitute \(b^2\) and \(c^2 = 100\) into the equation: \(a^2 + 25 = 100\).
• Solve for \(a^2\): \(a^2 = 75\).
• Take the square root of both sides: \(a = \sqrt{75} \approx 8.7\).

Rounding 8.7 to the nearest tenth gives us a complete set of triangle sides. This process showcases how foundational the Pythagorean Theorem is to right triangle trigonometry.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when one of the trigonometric ratios like sine, cosine, or tangent is known. They are the reverse operations of these trigonometric functions, allowing us to calculate an angle from a ratio.In this exercise, we used the inverse sine function, which is written as \( \sin^{-1} \) or arc\(\sin\). It takes a ratio as its input and gives the angle whose sine is that ratio. This is particularly useful in right triangles when you have the opposite side and the hypotenuse.Given that \( \sin(A) = \frac{5}{10} = 0.5 \), the inverse sine function helps us find \( \angle A \). By calculating \( \sin^{-1}(0.5) \), you achieve the angle for \(\angle A\) which is:

• \( \sin^{-1}(0.5) = 30^\circ \)

This simple inverse process gives you accurate angles, helping to solve the triangle without directly measuring it. It's a powerful way to fully understand the framework of right triangle trigonometry.

Angle Sum of a Triangle

The sum of all angles in any triangle is always 180 degrees. This is a fundamental rule in geometry that applies to all triangles, including right triangles. Knowing this, you can find unknown angles if you have at least two angle measures.For a right triangle, one angle is always 90 degrees. Using this knowledge simplifies finding the other angles. In the example given, we already know \( \angle C = 90^\circ \) and \(\angle A = 30^\circ \). To find \(\angle B\), use the angle sum property:

• Add the known angles: \(90^\circ + 30^\circ = 120^\circ\)
• Subtract this sum from 180 degrees: \(180^\circ - 120^\circ = 60^\circ\)

Therefore, \( \angle B = 60^\circ \).This simple subtraction confirms the remaining angle. Understanding this concept ensures you can always check your work against the total angle sum, maintaining accuracy in calculations and strengthening your grasp of triangle geometry.