Problem 37 - Algebra 2

Question

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(m \angle A=52^{\circ}, c=10\)

Step-by-Step Solution

Answer

\(\angle B = 38^\circ \), \(a \approx 7.9\), \(b \approx 6.4\)

Step 1: Find angle B

Because it's a triangle, the sum of the angles is 180 degrees. Since we already know that \( \angle A = 52^\circ \) and \( \angle C = 90^\circ \), we can subtract those from 180 to get \( \angle B \). So, \( m \angle B = 180^\circ - 90^\circ - 52^\circ = 38^\circ \).

Step 2: Find side a

In a right triangle, the side opposite the angle A, denoted 'a', can be computed using sine of the angle A and the length of hypotenuse (c). Thus, \( a = c \sin(A) = 10 \sin(52^\circ) \). After calculation, you'll find \( a \approx 7.9 \).

Step 3: Find side b

Similarly, the side opposite the angle B, denoted 'b', can be computed using sine of the angle B and the length of the hypotenuse (c). Thus, \( b = c \sin(B) = 10 \sin(38^\circ) \). After calculation, you'll find \( b \approx 6.4 \).

Key Concepts

Angle Sum Property

In any triangle, the sum of all interior angles is always 180 degrees. This key principle is known as the angle sum property. It helps us find unknown angles when other angle measures are known. For our problem,

• Triangle ABC is a right triangle with one angle, \(\angle C\), fixed at 90 degrees.
• Another angle, \(\angle A\), is given as 52 degrees.

To determine the remaining angle, \(\angle B\), we subtract the known angles from 180 degrees:
\[m \angle B = 180^\circ - 90^\circ - 52^\circ = 38^\circ.\]This method ensures the total is always 180 degrees, confirming our calculations are correct.

Sine Function

The sine function is a fundamental tool in trigonometry often used in right triangles. It's defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse. Here's how we used sine in our exercise:

• The sine of \(\angle A\), \(\sin(52^\circ)\), helps find side \(a\).
• Labeled as \(c\), the hypotenuse is known to be 10.

Using this information, we calculate:
\[a = c \sin(52^\circ) \approx 10 \times 0.7880 \approx 7.9.\]Similarly, for \(\angle B\):

• \(\sin(38^\circ)\) is used for finding side \(b\).

The calculation here is:
\[b = c \sin(38^\circ) \approx 10 \times 0.6157 \approx 6.4.\]Each calculation shows how the sine function can reveal missing side lengths in a right triangle.

Finding Missing Sides

Once the angles are determined and the hypotenuse is known, finding the missing sides in a right triangle becomes straightforward. We rely on trigonometric functions such as sine introduced earlier:

• The formula \(a = c \sin(\angle A)\) gives us side \(a\).
• Similarly, \(b = c \sin(\angle B)\) allows us to find side \(b\).

These equations simply require plugging in values for the angles and the hypotenuse. After calculating these:
- Side \(a\) was found to be approximately 7.9.- Side \(b\) turned out to be approximately 6.4.
These steps illustrate how trigonometric functions simplify solving for unknown sides in right triangles.

Angle Calculation

Calculating the unknown angles in our right triangle exercise involves understanding both the angle sum property and using right angle constraints. First, knowing \(\angle C\) as 90 degrees establishes the context of a right triangle. With \(\angle A\) provided,

• Calculate \(\angle B\) by subtracting the known angles from 180 degrees.

Using this process:
\[m \angle B = 180^\circ - (52^\circ + 90^\circ) = 38^\circ.\]
This calculation confirms the angle measurements advance our understanding of the entire triangle, ensuring all components (angles and sides) fit harmoniously together.