Problem 5 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) find the exact values of the remaining parts. $$a=5, \quad b=5$$

Step-by-Step Solution

Answer

\( c = 5\sqrt{2} \), \( \alpha = 45^\circ \), \( \beta = 45^\circ \).

Step 1: Identify the Right Triangle Properties

Since \( \gamma = 90^{\circ} \), triangle \( \triangle ABC \) is a right triangle with \( C \) being the right angle. The sides \( a \) and \( b \) are the legs of the triangle and \( c \) is the hypotenuse.

Step 2: Apply the Pythagorean Theorem

For any right triangle, the Pythagorean theorem states \( a^2 + b^2 = c^2 \). Since \( a = 5 \) and \( b = 5 \), substitute these values into the equation: \( 5^2 + 5^2 = c^2 \). Calculate \( c \).

Step 3: Solving for the Hypotenuse \( c \)

Calculate \( 5^2 + 5^2 = c^2 \). This gives \( 25 + 25 = c^2 \), so \( c^2 = 50 \). Solving for \( c \), we find \( c = \sqrt{50} = 5\sqrt{2} \).

Step 4: Calculate \( \alpha \) and \( \beta \) Using Trigonometric Ratios

Since both legs \( a \) and \( b \) are equal, the triangle is isosceles. In an isosceles right triangle, the angles opposite the equal sides are equal. Using the property \( \alpha + \beta + \gamma = 180^\circ \), \( \alpha + \beta = 90^\circ \), we find \( \alpha = \beta = 45^\circ \).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in right triangle trigonometry. It states that in any 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, called the legs. Mathematically, it is expressed as:\[ a^2 + b^2 = c^2 \]where:

• \(a\) and \(b\) are the lengths of the two legs,
• \(c\) is the length of the hypotenuse.

For example, in an isosceles right triangle where both legs are equal, say \(a = b = 5\), you can apply the Pythagorean Theorem. Here, substituting the values gives:\[ 5^2 + 5^2 = c^2 \]This simplifies to \(50 = c^2\), where solving for \(c\) yields \(c = \sqrt{50} = 5\sqrt{2}\). This demonstrates how the theorem helps us determine unknown sides of right triangles when other sides are known.

Isosceles Right Triangle

An isosceles right triangle is a specific type of right triangle where two of its sides are equal in length. This also means that the angles opposite those sides are equal. By definition:

• The two equal sides are known as the legs of the triangle.
• The angle between these equal sides is \(90^{\circ}\) .
• The two other angles in the triangle are \(45^{\circ}\) each.

In our original problem, the triangle had legs \(a\) and \(b\) both equal to \(5\), making it an isosceles right triangle. The symmetry of the triangle ensures that both non-right angles \(\alpha\) and \(\beta\) measure \(45^{\circ}\), while the Pythagorean Theorem confirms that the hypotenuse is \(5\sqrt{2}\). This property simplifies calculating angles and sides, as you already know the angles if you know two sides are equal.

Trigonometric Ratios

Trigonometric ratios are essential tools that relate the angles and sides of triangles. In right triangles, the primary trigonometric ratios are sine, cosine, and tangent. These are defined as follows:

• Sine (\(\sin\)): opposite leg over hypotenuse
• Cosine (\(\cos\)): adjacent leg over hypotenuse
• Tangent (\(\tan\)): opposite leg over adjacent leg

For the isosceles right triangle in our example:

• The sine and cosine of \(45^{\circ}\) are both equal to \( \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{2} \).
• The tangent of \(45^{\circ}\) is 1, as the opposite and adjacent sides are equal.

Understanding these ratios helps in calculating unknown sides or angles in right triangles. Particularly when solving for angles using sides, these ratios are invaluable.

Angle Calculation in Right Triangles

Angle calculation in right triangles is straightforward, thanks to the sum of angles in a triangle being \(180^{\circ}\). In any right triangle, this principle simplifies to:\[ \alpha + \beta + \gamma = 180^{\circ} \]Where \(\gamma\) is the right angle, thus:\[ \alpha + \beta = 90^{\circ} \]For an isosceles right triangle, where the other two angles are equal, each measures \(45^{\circ}\). Calculating angles is crucial when you're given some values and need the others. Since the triangle in our exercise is isosceles with equal legs, this inherent property saves time and steps. Knowing the equal split of the remaining \(90^{\circ}\) between two equal angles, the quick calculation leads directly to \(\alpha = \beta = 45^{\circ}\). This property of angles is used frequently in solving real-life problems involving right triangles.