Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-8: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), find the exact values of the remaining parts. $$ \beta=45^{\circ}, \quad b=35 $$

Step-by-Step Solution

Answer

\( a = 35, \ c = 35\sqrt{2} \); \( \alpha = 45^{\circ} \).

Step 1: Identify the given parts

In the right triangle \( \triangle ABC \), we are given: \( \gamma = 90^{\circ} \), \( \beta = 45^{\circ} \), and the side \( b = 35 \). We need to find the remaining angle \( \alpha \), and the sides \( a \) and \( c \).

Step 2: Find angle \( \alpha \)

Since the sum of angles in a triangle is \( 180^{\circ} \) and \( \gamma = 90^{\circ} \), the remaining two angles \( \alpha \) and \( \beta \) must sum to \( 90^{\circ} \). Given \( \beta = 45^{\circ} \), we have \( \alpha = 90^{\circ} - 45^{\circ} = 45^{\circ} \).

Step 3: Use trigonometric ratios to find side \( a \)

In a right triangle with \( \beta = 45^{\circ} \), the sides opposite the angles \( 45^{\circ} \) are equal if it's an isosceles right triangle, which it is here as both \( \alpha \) and \( \beta \) are \( 45^{\circ} \). Since \( b = 35 \), \( a = 35 \).

Step 4: Find hypotenuse \( c \) using Pythagorean theorem

Use the Pythagorean theorem, \( a^2 + b^2 = c^2 \), to find the hypotenuse. Substitute \( a = 35 \) and \( b = 35 \):\[ c^2 = 35^2 + 35^2 = 2 \times 35^2 = 2 \times 1225 = 2450 \]Taking the square root gives \( c = \sqrt{2450} = 35 \sqrt{2} \).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in right triangle trigonometry. It is used to relate the sides of a right triangle. The theorem states that for a right triangle with sides of length \( a \) and \( b \), and hypotenuse \( c \), the equation \( a^2 + b^2 = c^2 \) holds true.

This theorem helps us calculate the length of one side if the other two are known. For example, in the given problem, both \( a \) and \( b \) are equal to 35, allowing us to determine \( c \), the hypotenuse, using the theorem.

By substituting \( a \) and \( b \) into the equation, we find \( c^2 = 35^2 + 35^2 \). Solving that gives \( c = \sqrt{2450} = 35 \sqrt{2} \). Understanding and applying this theorem is crucial for solving problems involving right triangles.

Trigonometric Ratios

Trigonometric ratios are vital for understanding the relationships between the angles and sides of right triangles. The basic trigonometric ratios are sine, cosine, and tangent.

For a right triangle with an angle \( \theta \), these ratios are defined as:

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

In our exercise, however, we focus on using the angles to directly relate the sides since both angles \( \alpha \) and \( \beta \) are \( 45^{\circ} \).

In an isosceles right triangle, these two \( 45^{\circ} \) angles ensure the sides opposite these angles are equal. Thus, the trigonometric ratios can confirm findings but serve more as a supporting tool for verifying calculations when the angles are known.

Isosceles Right Triangle

An isosceles right triangle is a special form of a right triangle where the two legs are equal in length. This type of triangle is characterized by angles of \( 90^{\circ} \), \( 45^{\circ} \), and \( 45^{\circ} \).

In this exercise, understanding that the triangle is isosceles simplifies finding the side lengths. With both \( \alpha \) and \( \beta \) equal to \( 45^{\circ} \), the side across from these angles is the same, hence \( a = b = 35 \).

This also confirms that in an isosceles right triangle, the hypotenuse \( c \) is related to the equal legs through the relationship \( c = a \sqrt{2} \), which is consistent with our calculated hypotenuse of \( 35 \sqrt{2} \). Understanding these properties allows for quick assessments of side lengths whenever dealing with \( 45^{\circ} \) angles.