Problem 16 - Algebra and Trigonometry

Question

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=7, c=3, B=90^{\circ}$$

Step-by-Step Solution

Answer

The values are approximately: \(b \approx 6.3\), \(A \approx 48^{\circ}\), and \(C \approx 42^{\circ}\)

Step 1: Find length of side b

We apply the Pythagorean theorem \( a^2 = b^2 + c^2 \). Isolate \( b \) to obtain \( b = \sqrt{a^2 - c^2} \). Substituting \( a = 7 \) and \( c = 3 \) into the equation gives \( b = \sqrt{7^2 - 3^2} = \sqrt{40} \approx 6.3 \) to the nearest tenth.

Step 2: Find angle A

We use the tangent trigonometric ratios which is \(\tan(\theta) = \frac{opposite}{adjacent}\). Since angle A is opposite side a and adjacent to side b, the formula becomes \( \tan(A) = \frac{a}{b} \). Therefore \(A = \arctan(\frac{a}{b})\). Substituting \( a = 7 \) and \( b = 6.3 \) into the equation gives \( A = \arctan(\frac{7}{6.3})\), which is approximately \(48^{\circ}\) when rounded to the nearest degree.

Step 3: Find angle C

Because the sum of angles in a triangle equals \(180^{\circ}\), we subtract the known angles from \(180^{\circ}\) to find angle C. So, \( C = 180^{\circ} - A - B \). Substituting \( A = 48^{\circ} \) and \( B = 90^{\circ} \) into the equation gives \( C = 180^{\circ} - 48^{\circ} - 90^{\circ} = 42^{\circ}\).

Key Concepts

Pythagorean theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially useful in solving right triangles.
It establishes a relationship between the lengths of the sides of a right triangle.

• The theorem states: \[ a^2 = b^2 + c^2 \]where \(a\) is the hypotenuse (the longest side), and \(b\) and \(c\) are the other two sides.
• In our exercise, given \(a = 7\) and \(c = 3\), we need to find side \(b\).

By rearranging the formula, we derive:\[ b = \sqrt{a^2 - c^2} \]Substituting the known values gives:\[ b = \sqrt{7^2 - 3^2} = \sqrt{49 - 9} = \sqrt{40} \approx 6.3 \]This calculation shows how the Pythagorean Theorem helps us find a missing side in a right triangle.

tangent function

The tangent function is one of the primary trigonometric functions used to find angles and sides in right triangles.
It is specifically helpful when you know two sides of the triangle and need to find an angle.

• The tangent ratio is defined as:\[ \tan(\theta) = \frac{opposite}{adjacent} \]
• In the given triangle, angle \(A\) is opposite side \(a = 7\) and adjacent to side \(b = 6.3\).

To find angle \(A\), we use:\[ A = \arctan\left(\frac{a}{b}\right) = \arctan\left(\frac{7}{6.3}\right) \]Calculating this gives us\[ A \approx 48^{\circ} \]The tangent function thus provides us a practical tool to deduce angles from side lengths.

angle sum property

The angle sum property is a basic yet powerful tool for working with triangles.This property tells us that the sum of all interior angles in any triangle is always \(180^{\circ}\).
This holds true for all triangles, including right triangles.

• In a right triangle, one angle is always \(90^{\circ}\).
• To find the remaining angle \(C\), given angles \(A\) and \(B\), you use the formula:\[ C = 180^{\circ} - A - B \]

So with \(A = 48^{\circ}\) and \(B = 90^{\circ}\), we find:\[ C = 180^{\circ} - 48^{\circ} - 90^{\circ} = 42^{\circ} \]The angle sum property facilitates finding unknown angles simply by subtraction.

right angle

Right angles are central to understanding right triangles and trigonometry.A right angle measures exactly \(90^{\circ}\) and signifies perpendicularity between two lines or edges.
In a right triangle:

• One of the triangle's three angles is always a right angle, setting the framework for Pythagorean relations and trigonometric calculations.
• This right angle affects the two remaining angles, ensuring their measures sum up to \(90^{\circ}\). Hence, if one angle is found, the other can be quickly calculated using the angle sum property.

Understanding the concept of a right angle equips us to explore other mathematical ideas like altitude and slope.It is the foundational building block in solving problems involving right triangles.