Problem 11 - Technical Mathematics with Calculus

Question

Sketch each right triangle and find all missing parts. Work to three significant digits and express the angles in decimal degrees. $$a=382 \quad b=274$$

Step-by-Step Solution

Answer

Hypotenuse (c) \approx 470.11, one angle \theta \approx 35.685^\circ, and the other angle \approx 54.315^\circ.

Step 1: Sketch the Right Triangle

Draw a right triangle with sides labeled as 'a' and 'b', and the right angle between them. The side 'a' is opposite the right angle, and the side 'b' is adjacent to the right angle.

Step 2: Calculate the Hypotenuse (c)

Use the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides. \[c = \sqrt{a^2 + b^2}\]\[c = \sqrt{382^2 + 274^2}\]\[c = \sqrt{145924 + 75076}\]\[c = \sqrt{221000}\]\[c \approx 470.11\]

Step 3: Calculate the Angles

Use trigonometric ratios to find the angles. For the angle adjacent to side 'a', use the tangent function where \[\tan(\theta) = \frac{opposite}{adjacent}\]\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]\[\theta = \tan^{-1}\left(\frac{274}{382}\right)\]\[\theta \approx 35.685^\circ\]The angle opposite side 'b' is given by \[90^\circ - \theta \approx 54.315^\circ\], because the angles in a right triangle must sum to 90 degrees, not including the right angle itself.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, specifically when dealing with right triangles. It asserts that for any right-angled triangle, the square of the length of the hypotenuse (which is the side opposite the right angle) is equal to the sum of the squares of the lengths of the triangle's other two sides.

The mathematical expression of this theorem is \[ c^2 = a^2 + b^2 \] where 'c' is the hypotenuse, while 'a' and 'b' are the other two sides of the triangle. This theorem allows us to calculate the length of any side of a right triangle if we know the lengths of the other two sides. It is a key tool in many fields, including architecture, engineering, and even navigation.

Understanding the Pythagorean theorem and its uses is crucial for solving problems related to right triangles, like finding the distance between two points on a coordinate plane, or determining the reach of a ladder leaning against a wall.

Trigonometric Ratios

Trigonometric ratios are used to relate the angles of a triangle to the lengths of its sides. These ratios are particularly useful in right triangles. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).

• The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

These ratios can be remembered by the acronym SOH-CAH-TOA. In practical terms, trigonometric ratios allow us to find unknown angles or side lengths in right triangles when given certain measurements. For instance, if we know the lengths of two sides, we can determine the measure of the angles by using inverse trigonometric functions.

Hypotenuse Calculation

The hypotenuse is the longest side of a right triangle, opposite the right angle. Calculating the hypotenuse is straightforward when you have the lengths of the other two sides, thanks to the Pythagorean theorem. By rearranging the expression \[ c^2 = a^2 + b^2 \] we can solve for 'c', the hypotenuse, which yields \[ c = \sqrt{a^2 + b^2} \].

In real-world problems, the hypotenuse often represents the 'direct distance' between two points. It's commonly encountered in situations like determining the width of a river without physically crossing it, or figuring out the straight-line flight distance between two locations. Accurate hypotenuse calculation is crucial for various applications, from construction to navigation.

Angle Measurement

Angle measurement in right triangles is essential for understanding the relationships between the sides and angles. Once the lengths of the sides are known, angles can be calculated using the trigonometric ratios mentioned earlier. Given one acute angle, it is possible to find the other, since the sum of the angles in a triangle is always 180 degrees, and the right angle accounts for 90 of those degrees.

Angles are commonly measured in degrees (°), and in some contexts, radians (rad) are used. In the example provided, the use of inverse tangent, denoted as \(tan^{-1}\), allowed us to calculate an acute angle given the lengths of the opposite and adjacent sides. Subsequently, subtracting this angle from 90° gives us the measure of the other acute angle. This process enables us to fully define the shape of any right triangle and is the foundation for many practical applications, such as land surveying, construction, and creating accurate maps.