Problem 10 - Precalculus

Question

In isosceles triangle \(A B C\), the sides are of length \(A C=B C=8\) and \(A B=4 .\) Find the angles of the triangle. Express the answers both in radians, rounded to two decimal places, and in degrees, rounded to one decimal place. Hints: To find \(\angle A,\) start by drawing an altitude from \(C\) to side \(\overline{A B}\). Then for \(\angle C\), use the fact that the sum of the angles in a triangle is \(\pi\) radians or \(180^{\circ}\).

Step-by-Step Solution

Answer

\(\angle A = 1.32\,\text{rad}, 75.5^\circ\) and \(\angle C = 0.51\,\text{rad}, 28.9^\circ\).

Step 1: Draw the Altitude and Divide the Triangle

In isosceles triangle \(ABC\), draw an altitude from vertex \(C\) to side \(AB\). This altitude will divide the triangle into two right triangles, \(ACD\) and \(BCD\), where \(D\) is the point on \(AB\) intersected by the altitude. Since \(AB = 4\), each segment \(AD\) and \(BD\) will be \(2\).

Step 2: Use the Pythagorean Theorem to Find \(CD\)

In right triangle \(ACD\), use the Pythagorean theorem. We know \(AC = 8\) and \(AD = 2\). The equation is:\[ AC^2 = AD^2 + CD^2 \]\[ 8^2 = 2^2 + CD^2 \]\[ 64 = 4 + CD^2 \]\[ CD^2 = 60 \]\[ CD = \sqrt{60} = 2\sqrt{15} \]

Step 3: Use Trigonometry to Find \(\angle A\)

In triangle \(ACD\), use trigonometric ratios to find \(\angle A\). We have:\[ \cos(\angle A) = \frac{AD}{AC} = \frac{2}{8} = \frac{1}{4} \]Use the inverse cosine function:\[ \angle A = \cos^{-1}(\frac{1}{4}) \approx 1.318 \text{ radians} \approx 75.5^\circ \]

Step 4: Calculate \(\angle C\) Using the Total Angle Sum

The sum of angles in a triangle is \(\pi\) radians or \(180^\circ\). Since triangle \(ABC\) is isosceles, \(\angle B = \angle A\). Therefore, the equation is:\[ 2\angle A + \angle C = \pi \]\[ 2(1.318) + \angle C = \pi \]\[ \angle C = \pi - 2(1.318) \approx 0.505 \text{ radians} \]

Step 5: Convert \(\angle C\) to Degrees

To find \(\angle C\) in degrees, convert from radians:\[ \angle C \approx 0.505 \times \frac{180}{\pi} \approx 28.9^\circ \]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right-angled 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 basic formula is:\[ c^2 = a^2 + b^2 \]where:

• \( c \): Length of the hypotenuse
• \( a \) and \( b \): Lengths of the other two sides

In the context of the isosceles triangle problem, we use this theorem to find the length of the altitude (\(CD\)) from vertex \(C\) to side \(AB\). The theorem allows us to split the triangle into two right triangles, making it possible to apply the formula to find unknown side lengths. With known values of the base (\(AD=2\)) and side (\(AC=8\)), the calculation\(CD = \sqrt{60} = 2\sqrt{15}\)proceeds effortlessly by solving for \( CD \) using \( AC^2 = AD^2 + CD^2 \). This process highlights the Pythagorean Theorem's power in solving geometric problems involving right triangles.

Trigonometry

Trigonometry connects angles and side lengths in right triangles, using functions like sine, cosine, and tangent. These trigonometric ratios are invaluable in solving for unknown attributes of a triangle when certain values are known. For instance, to determine \( \angle A\) in triangle \( ACD \), we use the cosine function because the adjacent side (\( AD \)) and the hypotenuse (\( AC \)) are known.The formula used is:\[ \cos(\angle A) = \frac{AD}{AC} \]Substituting the known side lengths gives:\[ \cos(\angle A) = \frac{2}{8} = \frac{1}{4} \]Then, using the inverse cosine function, we find:\[ \angle A = \cos^{-1}\left(\frac{1}{4}\right) \approx 1.318 \text{ radians} \approx 75.5^\circ \]This calculation displays how trigonometric ratios provide a straightforward method to find angles in right triangles when side lengths are known.

Angle Calculation

Calculating angles in a triangle involves understanding the properties of triangles and using known sides or angles effectively. In any triangle, the sum of the interior angles is always \( 180^\circ \) or \( \pi \) radians. This property is particularly useful when calculating unknown angles if at least two angles are known.In the isosceles triangle problem, we know \( \angle A = \angle B \) due to the triangle's symmetry. This allows us to write the equation combining \( \angle A \) and \( \angle C \):\[ 2\angle A + \angle C = \pi \]The angle \( \angle A \) has already been calculated using trigonometry:\[ 2 \times 1.318 + \angle C = \pi \]Solving for \( \angle C \), we find:\[ \angle C = \pi - 2 \times 1.318 \approx 0.505 \text{ radians} \]This equation shows the elegance of angle calculation in using simple relationships to deduce unknown angles.

Radians and Degrees Conversion

Radians and degrees are two units used to measure angles, with radians being the standard unit in mathematics. The relationship between them is:\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]This conversion enables us to switch between radians and degrees, depending on which unit is more useful in a given context. Often, working in degrees is familiar and easier to visualize, whereas radians are more natural when applied in calculus and trigonometry.In the problem at hand, after finding \( \angle C \) in radians:\[ \angle C \approx 0.505 \text{ radians} \]Use the conversion formula:\[ \angle C \times \frac{180}{\pi} \approx 28.9^\circ \]Being comfortable with this conversion is essential as it allows for seamless communication of angle measurements in both everyday situations and more advanced mathematical calculations.