Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(a=958\) meters; \(b=489\) meters

Step-by-Step Solution

Answer

\( c \approx 1075.32 \) meters, \( A \approx 62^{\circ} 30' \), \( B \approx 27^{\circ} 30' \).

Step 1: Identify the Given Values

We are given a right triangle where \( a = 958 \) meters and \( b = 489 \) meters, and angle \( C = 90^{\circ} \). We need to find side \( c \) and angles \( A \) and \( B \).

Step 2: Use the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, \( a^2 + b^2 = c^2 \). Substituting the given values, we have:\[ c^2 = 958^2 + 489^2 \]Calculate \( c \) by evaluating the expression.

Step 3: Calculate Hypotenuse c

Compute the squares and their sum:\[ c^2 = 958^2 + 489^2 = 917764 + 239121 = 1156885 \]Then, take the square root to find \( c \):\[ c = \sqrt{1156885} \approx 1075.32 \text{ meters} \]

Step 4: Find Angle A Using Trigonometric Ratios

Use the tangent function, \( \tan(A) = \frac{a}{b} \), to find angle \( A \).\[ \tan(A) = \frac{958}{489} \approx 1.959 \]Then, use the arctan function:\[ A \approx \arctan(1.959) \approx 62^{\circ} 30' \]

Step 5: Find Angle B

Angles in a triangle sum to \( 180^{\circ} \). Using this, \( B \) can be found as:\[ B = 180^{\circ} - 90^{\circ} - A \]Substitute \( A \):\[ B = 180^{\circ} - 90^{\circ} - 62^{\circ} 30' = 27^{\circ} 30' \]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is fundamental in solving right triangles. This theorem suggests that, in a 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 other two sides. It can be expressed as: \[ a^2 + b^2 = c^2 \] In our given exercise, we know two sides: \( a = 958 \) meters and \( b = 489 \) meters. We can find the hypotenuse \( c \) using the formula:

• Calculate \( a^2 \) and \( b^2 \).
• Add these values together to get \( c^2 \).
• Finally, take the square root of \( c^2 \) to find \( c \).

The result is the length of the hypotenuse, which helps us fully define the triangle.

Trigonometric Ratios

Trigonometric ratios are used to calculate the unknown angles in a right triangle, given the lengths of its sides. These ratios involve three primary functions: sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function relates to the angles and sides of the right triangle. In this exercise, we focus on the tangent for angle \( A \), which is given by the ratio of the opposite side to the adjacent side: \[ \tan(A) = \frac{a}{b} \] For our given sides, \( a = 958 \) meters and \( b = 489 \) meters, we calculate \( \tan(A) \). Using this value, the arctan function is applied, allowing us to determine angle \( A \). Trigonometric ratios provide a robust way to transition from side lengths to angles, thereby solving the triangle entirely.

Angle Calculation

The sum of angles in any triangle is always \( 180^{\circ} \). In a right triangle, one angle is \( 90^{\circ} \), leaving the sum of the other two angles to be \( 90^{\circ} \). Once one angle is known, the other can be easily calculated using subtraction. For angle \( B \), once we determine angle \( A \) to be roughly \( 62^{\circ} 30' \), we subtract from \( 90^{\circ} \):

• Subtract angle \( A \) from \( 90^{\circ} \)
• Calculate angle \( B \)

Thus, if angle \( A = 62^{\circ} 30' \), then angle \( B = 27^{\circ} 30' \). Understanding this simple measure of angles is key to solving right triangles.

Degree and Minute Conversion

Degrees and minutes are a way to express angles more precisely. One degree is equivalent to 60 minutes. So, when an angle is calculated in decimal degrees, it may need to be converted to degrees and minutes for a more precise representation. Suppose you have an angle in decimal form, say \( 62.5^{\circ} \). To convert this, follow these steps:

• Take the whole number as the degree (\( 62^{\circ} \)).
• Multiply the decimal (.5) by 60 to get the minutes (\( 30' \)).

This means \( 62.5^{\circ} \) is equal to \( 62^{\circ} 30' \). By understanding and using this conversion properly, angle measures can maintain high precision, making them useful in various applications.