Problem 39 - 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. \(m \angle B=17.2^{\circ}, b=8.3\)

Step-by-Step Solution

Answer

Side a is approximately 7.9 units, side c is approximately 11.1 units, and the measure of angle A is approximately 72.8 degrees.

Step 1: Determine the Length of Side a

Since \triangle ABC is a right-angled triangle with \theta B known, use the trigonometric relation for cosine to determine the length of side a. The cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse: \[ \text{cos}(B) = \frac{a}{b}\] \[\text{cos}(17.2^\text{o}) = \frac{a}{8.3}\]. Multiply both sides by 8.3 to find the length of side a: \[ a = 8.3 \times \text{cos}(17.2^\text{o}).\]

Step 2: Calculate the Length of Side c

Using the Pythagorean theorem for right-angled triangles (where c is the hypotenuse), we can calculate the length of side c: \[ a^2 + b^2 = c^2\]. We already calculated the value of a from the previous step; substitute the values of a and b into the equation: \[ c = \big(a^2 + 8.3^2\big)^{1/2}.\]

Step 3: Determine the Measure of Angle A

To find angle A, understand that in a right-angled triangle, the sum of the angles is 180 degrees. Angle A can be found by subtracting the right angle and angle B from 180 degrees: \[ m\triangle A = 180^\text{o} - m\triangle C - m\triangle B = 180^\text{o} - 90^\text{o} - 17.2^\text{o}. \]

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in the field of trigonometry, especially when it comes to right triangles. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can be expressed through the formula: \[a^2 + b^2 = c^2\].

The hypotenuse is typically denoted by 'c,' while the other two sides are represented by 'a' and 'b.' It’s important for students to recognize the right angle, as this theorem only applies to right-angled triangles. In practice, if we know the lengths of any two sides of a right triangle, we can always solve for the third side using this relationship. This makes the Pythagorean theorem a powerful tool for understanding the geometry of right triangles and is essential for solving many types of geometric problems.

Cosine Function

The cosine function is one of the primary trigonometric functions used to relate the angles of a right triangle to the lengths of its sides. In the context of a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. The formula is expressed as: \[ \text{cos}(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \].

In your exercise, this ratio allowed us to find the length of side 'a' when the angle at 'B' and the length of side 'b' (opposite 'B') were known. With side 'b' serving as the hypotenuse in this scenario, it becomes a practical application of the cosine function. By using a calculator or a trigonometric table, we can easily find the cosine value for any given angle, which is key when solving for missing sides of a triangle.

Trigonometric Ratios

Trigonometric ratios are the basis of trigonometry and are used to relate the angles of a triangle to the ratios of its sides. There are six fundamental trigonometric functions, but in the context of a right triangle, we primarily use the sine, cosine, and tangent functions. These ratios are defined as follows:

• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.

Understanding these ratios is critical when attempting to find unknown sides or angles in right triangles. They create a link between geometry and algebra, allowing for the solution of triangles that would otherwise be difficult to compute.

Calculating Angles

Calculating angles in right triangles is an essential skill in trigonometry, and it requires an understanding of the trigonometric ratios. When you have at least one angle (other than the right angle) and one side of a right triangle, you can calculate the other angles and sides. The process typically involves:

• Using the trigonometric ratios to find the missing sides when one angle and one side are known.
• Applying the fact that the sum of the angles in any triangle is equal to 180 degrees to find the missing angles.

In the given exercise, angle A could be found by subtracting the known angles, B and the right angle at C, from the total 180 degrees: \[ m\angle A = 180^\circ - 90^\circ - 17.2^\circ \]. This calculation is straightforward and uses the fact that a triangle's angles must sum up to 180 degrees—a key concept when calculating angles in any triangle.