Problem 2 - 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. $$\beta=45^{\circ}, \quad b=35$$

Step-by-Step Solution

Answer

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

Step 1: Identify Right Triangle

Since \(\gamma = 90^{\circ}\), we have a right-angled triangle. Knowing one angle is \(90^{\circ}\), the sum of the angles in a triangle being \(180^{\circ}\) allows us to find the third angle given \(\beta = 45^{\circ}\).

Step 2: Calculate the Missing Angle

In right triangle \(ABC\), the angles sum to \(180^{\circ}\). Therefore, \(\alpha = 180^{\circ} - 90^{\circ} - 45^{\circ} = 45^{\circ}\). This means \(\alpha = 45^{\circ}\).

Step 3: Recognize a 45-45-90 Triangle

Given the angles \(45^{\circ}, 45^{\circ},\) and \(90^{\circ}\), triangle \(ABC\) is an isosceles right triangle (45-45-90 triangle), which implies that sides \(a\) and \(b\) are equal.

Step 4: Calculate the Hypotenuse

In a 45-45-90 triangle, the hypotenuse \(c\) is \(\sqrt{2}\) times larger than either leg. Therefore, \(c = b\sqrt{2} = 35\sqrt{2}\).

Step 5: Calculate Side a

Since \(a = b\) due to the properties of a 45-45-90 triangle, we find \(a = 35\).

Key Concepts

45-45-90 triangle

A 45-45-90 triangle is a special type of right triangle. It is known for having two angles that are each 45 degrees and one angle that is 90 degrees. This means that it is both a right triangle and a type of isosceles triangle. In these triangles, the legs, or the two equal sides, are congruent. The hypotenuse, which is the side opposite the 90-degree angle, is a special function of the leg length. This simplification comes from the Pythagorean theorem and the properties of squares. The calculation for the hypotenuse \(c\) given a leg \(b\) (or \(a\) since they are equal) in a 45-45-90 triangle is \(c = b\sqrt{2}\). This ratio helps solve various geometry problems efficiently by making calculations much simpler. For example, if one leg measures 35 units, then the hypotenuse would measure \(35\sqrt{2}\) units.

isosceles right triangle

Isosceles right triangles, such as the 45-45-90 triangle, combine properties from both isosceles and right triangles.

• An isosceles triangle has at least two equal sides.
• A right triangle features one angle that is 90 degrees.

In the isosceles right triangle, the two equal sides are the legs, and the right angle is formed between them. This unique combination equates the legs both in length and allows the hypotenuse's length to be straightforwardly expressed as a function of these legs. Typically, the special symmetry and proportional nature of this type of triangle make many geometric calculations more intuitive and straightforward. Knowing one side length can easily lead you to find the missing elements due to predictable ratios and equal measurements.

trigonometric identities

Trigonometric identities are fundamental relationships in trigonometry that relate the angles and sides of triangles. For right triangles, these identities are especially useful for understanding and calculating various side lengths and angle measures.

• For a 45-degree angle in a 45-45-90 triangle, the sine, cosine, and tangent are particularly easy to remember:
  • \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)
  • \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)
  • \(\tan(45^\circ) = 1\)

These identities help solve problems by creating formulas and equations when dealing with geometry in right triangles like our 45-45-90 case. When sides are known, these identities also help in determining the exact trigonometric ratios, which further allows for easy calculation of other triangle properties.