Problem 19 - Precalculus

Question

Solve each triangle. $$a=12, b=5, c=13$$

Step-by-Step Solution

Answer

It's a right triangle with angles approximately 67.38°, 22.62°, and 90°.

Step 1: Verify the triangle using the Pythagorean Theorem

Given the sides of the triangle as \( a = 12 \), \( b = 5 \), and \( c = 13 \), first check if it is a right triangle. According to the Pythagorean Theorem, \( c^2 = a^2 + b^2 \). Compute: \( 13^2 = 169 \) and \( 12^2 + 5^2 = 144 + 25 = 169 \). Since these are equal, the triangle is a right triangle with \( c \) as the hypotenuse.

Step 2: Find angle A using the sine function

Since it's a right triangle with \( c \) as the hypotenuse, use the sine function to find \( \angle A \). Use \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13} \). Calculate \( A = \arcsin\left(\frac{12}{13}\right)\). Using a calculator, \( A \approx 67.38^\circ \).

Step 3: Find angle B using the sine function

Since it's a right triangle, use the sine function to find \( \angle B \) given that \( b = 5 \). Use \( \sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13} \). Calculate \( B = \arcsin\left(\frac{5}{13}\right)\). Using a calculator, \( B \approx 22.62^\circ \).

Step 4: Determine angle C

Since this is a right triangle, \( \angle C \) is the right angle. Therefore, \( \angle C = 90^\circ \).

Step 5: Summarize the triangle

The triangle is solved with sides \( a = 12 \), \( b = 5 \), \( c = 13 \) and angles \( \angle A \approx 67.38^\circ \), \( \angle B \approx 22.62^\circ \), and \( \angle C = 90^\circ \).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry that helps determine whether a triangle is a right triangle. A right triangle has a special feature: one of its angles is exactly 90 degrees. This theorem states that in a triangle with sides labeled as \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse, the relationship is given by the equation \(c^2 = a^2 + b^2\).
This relation means that the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)). This is an effective way to verify if a triangle is right-angled.

• Example: For a triangle with sides \(a = 12\), \(b = 5\), and \(c = 13\), compute \(13^2 = 169\) and \(12^2 + 5^2 = 144 + 25 = 169\).
• Since these are equal, the triangle is a right triangle.

Using the Pythagorean theorem is essential for confirming the nature of a triangle and provides a strong basis for calculating other parameters like angles.

sine function

The sine function is one of the basic trigonometric functions. It relates a given angle in a triangle to the ratio of the length of the side opposite that angle to the length of the triangle’s hypotenuse.
In a right triangle, if you want to calculate an angle, and you know the length of the opposite side and the hypotenuse, you can use the sine function. The formula for sine is \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
For instance, if we need to find angle \(A\) where \(c\) is the hypotenuse, the formula would be \(\sin A = \frac{12}{13}\).

• Calculate \(A = \text{arcsin}\left(\frac{12}{13}\right)\)—which approximately equals \(67.38^\circ\).
• Similarly, for angle \(B\), \(\sin B = \frac{5}{13}\) results in \(B = \text{arcsin}\left(\frac{5}{13}\right)\)—which is about \(22.62^\circ\).

Thus, the sine function is highly useful when dealing with right triangles and helps to determine angles accurately.

triangle angles

A triangle's angles are an essential aspect of its geometry, and all triangles have specific characteristics regarding their angles. In any triangle, the sum of the interior angles is always \(180^\circ\). For right triangles, this rule allows for easier calculations because one angle is always \(90^\circ\).
Knowing this, if you calculate two angles of a triangle, the third can be deduced easily. For instance, with angles \(A\) and \(B\) known in a right triangle:

• Angle \(C\) is already known to be \(90^\circ\).
• If \(A \approx 67.38^\circ\) and \(B \approx 22.62^\circ\), adding \(A\) and \(B\) gives \(90^\circ\).

Such properties are crucial because they simplify the problem of solving triangles and help verify the consistency of calculations. When considering any right triangle, angle computations serve as a key part of understanding the complete shape and its properties.