Problem 10 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) approximate the remaining parts. $$\beta=64^{\circ} 20^{\prime}, \quad a=20.1$$

Step-by-Step Solution

Answer

\(\alpha = 25^{\circ} 40^{\prime}\); \(b \approx 9.14\); \(c \approx 22.38\).

Step 1: Identify Known Values

In triangle \(ABC\), we know that \(\gamma = 90^{\circ}\) (since it is a right triangle), \(\beta = 64^{\circ} 20^{\prime}\), and the side \(a = 20.1\). We need to find the other angle \(\alpha\), and the remaining sides \(b\) and \(c\).

Step 2: Calculate Angle \(\alpha\)

Since the sum of angles in any triangle is \(180^{\circ}\), we can find \(\alpha\) by using the equation: \[\alpha = 180^{\circ} - \beta - \gamma.\]Substitute the known values:\[\alpha = 180^{\circ} - 64^{\circ} 20^{\prime} - 90^{\circ} = 25^{\circ} 40^{\prime}.\]

Step 3: Calculate Side \(b\) Using Trigonometry

Using the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle, we have:\[\tan(\beta) = \frac{a}{b}.\]Solving for \(b\):\[b = \frac{a}{\tan(\beta)}.\]Substitute \(a = 20.1\) and \(\beta = 64^{\circ} 20^{\prime}\):\[b \approx \frac{20.1}{\tan(64^{\circ} 20^{\prime})} \approx 9.14.\]

Step 4: Calculate Hypotenuse \(c\) Using Trigonometry

We use the sine function, which is the ratio of the opposite side to the hypotenuse:\[\sin(\beta) = \frac{a}{c}.\]Solving for \(c\):\[c = \frac{a}{\sin(\beta)}.\]Substitute \(a = 20.1\) and \(\beta = 64^{\circ} 20^{\prime}\):\[c \approx \frac{20.1}{\sin(64^{\circ} 20^{\prime})} \approx 22.38.\]

Step 5: Verification Step: Check Calculations

Double-check calculations by computing the cosine of \(\beta\) and verifying side \(b\) using:\[\cos(\beta) = \frac{b}{c}.\]Calculate \(\cos(64^{\circ} 20^{\prime})\) and compare \(b\):\[\cos(64^{\circ} 20^{\prime}) \approx 0.435\quad \Rightarrow \quad b \approx c \times 0.435 \approx 9.14.\]The values are consistent, ensuring calculations are correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in understanding right triangles. When dealing with a right triangle, the main functions used are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions help relate the angles to the sides of the triangle.

• Sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse.
• Cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent of an angle is the ratio of the length of the side opposite the angle to the adjacent side.

These functions make it easy to solve for unknown sides and angles in a triangle once some parts are known. In our exercise, from \(\beta = 64^{\circ} 20^{\prime}\), the tangent function was used to find side \(b\) and the sine function was used to find the hypotenuse \(c\). These calculations are straightforward when you substitute known values into the function equations.

Angle Sum Property

The angle sum property is a fundamental concept in trigonometry and geometry. It states that the sum of the angles in any triangle is always \(180^{\circ}\). This is particularly helpful in right triangles where one angle is always \(90^{\circ}\).
To find an unknown angle in a triangle, simply subtract the known angles from \(180^{\circ}\). For example, if you have a right triangle with the angles \(\gamma = 90^{\circ}\) and \(\beta = 64^{\circ} 20^{\prime}\), you can find the third angle \(\alpha\) by calculating:\[\alpha = 180^{\circ} - 90^{\circ} - 64^{\circ} 20^{\prime}\]which equals \(25^{\circ} 40^{\prime}\). This property ensures you can always determine one unknown angle if the other two are known.

Triangle Side Lengths

Understanding side lengths in a right triangle involves using trigonometric relationships. With a right triangle, we have the hypotenuse, opposite side, and adjacent side. Depending on the known values, you can calculate the missing sides using trigonometric functions like sine, cosine, and tangent.

• Use the tangent function when you know an angle and one side, solving for the opposite or adjacent side.
• Employ the sine function to find the hypotenuse or opposite when you have an angle and another side.

In the example provided, side \(b\) was calculated using the tangent function:\[b = \frac{a}{\tan(\beta)}\]and hypotenuse \(c\) was calculated using the sine function:\[c = \frac{a}{\sin(\beta)}\]These formulas help break down complex problems into smaller, easier-to-solve parts, giving you clear methods to find any unknown side in a right triangle.