Problem 15 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) approximate the remaining parts. $$c=5.8, \quad b=2.1$$

Step-by-Step Solution

Answer

Approximate values are: \( a \approx 5.41 \), \( \alpha \approx 68.75^{\circ} \), \( \beta \approx 21.25^{\circ} \).

Step 1: Identifying Known Values

In the given right triangle, we have \ \( \gamma = 90^{\circ} \), the hypotenuse \( c = 5.8 \) and the side \( b = 2.1 \). We'll use these values to find the missing side \( a \) and the other two angles \( \alpha \) and \( \beta \).

Step 2: Using Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, \( a^2 + b^2 = c^2 \). Substituting the known values: \ \[ a^2 + (2.1)^2 = (5.8)^2 \] Simplify and solve for \( a^2 \): \ \[ a^2 + 4.41 = 33.64 \] \ \[ a^2 = 33.64 - 4.41 \] \ \[ a^2 = 29.23 \] \ \( a = \sqrt{29.23} \approx 5.41 \).

Step 3: Finding Angle \( \beta \) Using Sine Function

With \( \gamma = 90^{\circ} \), we use \( \sin \beta = \frac{b}{c} \). Thus, \ \[ \sin \beta = \frac{2.1}{5.8} \approx 0.362 \] \ Now, use the inverse sine function: \ \[ \beta = \sin^{-1}(0.362) \approx 21.25^{\circ} \]

Step 4: Finding Angle \( \alpha \) Using Angle Sum Property

In any triangle, the sum of the angles is \( 180^{\circ} \). For this right triangle, we have: \ \[ \alpha + \beta + 90^{\circ} = 180^{\circ} \] \ \[ \alpha = 180^{\circ} - 90^{\circ} - 21.25^{\circ} \approx 68.75^{\circ} \]

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in trigonometry, especially when dealing with right triangles. Its formula is expressed as \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse. In a right triangle, the hypotenuse is always the longest side and opposite the right angle.
This theorem helps us find the length of one side when we know the lengths of the other two. For example, if we have the values \( b = 2.1 \) and \( c = 5.8 \), we can find the missing side \( a \) by rearranging the formula to \( a^2 = c^2 - b^2 \).

• Calculate the square of the known sides.
• Subtract the square of side \( b \) from the square of the hypotenuse, \( c \).
• Finally, take the square root to find \( a \).

This simple and powerful theorem lays the groundwork for much of trigonometry, making it an essential tool in mathematics.

right triangles

Right triangles are a specific type of triangle that have one angle measuring exactly \( 90^{\circ} \). This right angle creates two legs and a hypotenuse, which is the side opposite the right angle. Understanding the properties of right triangles is crucial since they are used extensively in trigonometry and geometry.
With a right triangle:

• You can always apply the Pythagorean theorem.
• The hypotenuse is always the longest side.
• Two angles can be solved once one angle is known, using complementary angles to the right angle.

Right triangles serve as building blocks for more complex trigonometric problems, helping us understand angles, lengths, and their relationships.

sine function

The sine function is a trigonometric function that helps relate angles to the length of sides in right triangles. The sine of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the hypotenuse. In formula terms, it is written as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
In the context of right triangles:

• You can find angles when you know the lengths of the sides using \( \sin^{-1} \), the inverse sine function.
• For \( \beta \) in a right triangle, \( \sin \beta = \frac{b}{c} \).

The sine function is widely used in trigonometry to determine unknown angles or calculate side lengths in various applications, making it a fundamental aspect of angle-side relationships.

angle sum property

The angle sum property is an important characteristic of triangles which states that the sum of the interior angles in any triangle is always \( 180^{\circ} \). This property is particularly useful in solving for unknown angles in right triangles.
Here's how you can use it:

• In a right triangle, one angle is always \( 90^{\circ} \).
• The sum of the other two angles must be \( 90^{\circ} \) since \( \alpha + \beta + 90^{\circ} = 180^{\circ} \).
• If you know one non-right angle, you can easily find the other by subtracting from \( 90^{\circ} \).

This property is a simple yet powerful tool for solving triangles, allowing for the determination of angles with minimal information.