Problem 131 - Calculus Volume 1

Question

For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. $$a=85.3, b=125.5$$

Step-by-Step Solution

Answer

The length of side c is approximately 151.7. The six trigonometric functions are: \( \sin A \approx 0.56 \), \( \cos A \approx 0.83 \), \( \tan A \approx 0.68 \), \( \csc A \approx 1.79 \), \( \sec A \approx 1.20 \), \( \cot A \approx 1.47 \).

Step 1: Identify the Known Values

We have a right triangle ABC where \(a\) is the side opposite to angle A, \(b\) is the side adjacent to angle A, and \(c\) is the hypotenuse, which is unknown. The given values are \(a = 85.3\) and \(b = 125.5\).

Step 2: Use the Pythagorean Theorem

Since the triangle is a right triangle, we can apply the Pythagorean theorem. The theorem states: \[ c^2 = a^2 + b^2 \]Substitute the known values into the formula:\[ c^2 = (85.3)^2 + (125.5)^2 \]

Step 3: Perform the Calculation

Perform the calculations for each squared term:\[ c^2 = 7270.09 + 15750.25 = 23020.34 \] Now, take the square root of both sides to solve for \(c\):\[ c = \sqrt{23020.34} \approx 151.7 \] Thus, the length of the hypotenuse \(c\) is approximately 151.7.

Step 4: Calculate \( \sin A \)

The sine of angle A is the ratio of the opposite side to the hypotenuse:\[ \sin A = \frac{a}{c} = \frac{85.3}{151.7} \approx 0.56 \]

Step 5: Calculate \( \cos A \)

The cosine of angle A is the ratio of the adjacent side to the hypotenuse:\[ \cos A = \frac{b}{c} = \frac{125.5}{151.7} \approx 0.83 \]

Step 6: Calculate \( \tan A \)

The tangent of angle A is the ratio of the opposite side to the adjacent side:\[ \tan A = \frac{a}{b} = \frac{85.3}{125.5} \approx 0.68 \]

Step 7: Calculate \( \csc A \)

The cosecant of angle A is the reciprocal of sine A:\[ \csc A = \frac{1}{\sin A} \approx \frac{1}{0.56} \approx 1.79 \]

Step 8: Calculate \( \sec A \)

The secant of angle A is the reciprocal of cosine A:\[ \sec A = \frac{1}{\cos A} \approx \frac{1}{0.83} \approx 1.20 \]

Step 9: Calculate \( \cot A \)

The cotangent of angle A is the reciprocal of tangent A:\[ \cot A = \frac{1}{\tan A} \approx \frac{1}{0.68} \approx 1.47 \]

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding the relationships within a right triangle. There are six primary trigonometric functions, each relating to the angles and sides of a triangle:

• Sine (\(\sin\))
• Cosine (\(\cos\))
• Tangent (\(\tan\))
• Cosecant (\(\csc\))
• Secant (\(\sec\))
• Cotangent (\(\cot\))

The first three functions are the most commonly used and relate directly to the angles of a triangle. For example, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse. Similarly, cosine compares the adjacent side to the hypotenuse, while tangent is the ratio of the opposite side to the adjacent side.
The other three functions - cosecant, secant, and cotangent - are the reciprocals of sine, cosine, and tangent, respectively. Understanding these functions helps us solve for unknown sides and angles in right triangles, making them crucial in fields like engineering and physics.
Together, they form a foundation for more complex trigonometric calculations and are particularly valuable when working with periodic functions in mathematics.

Pythagorean Theorem

The Pythagorean theorem is a timeless principle in geometry that relates the sides of a right triangle. It is a specific formula that applies only to right triangles, where one angle is exactly 90 degrees. The theorem states:\[ c^2 = a^2 + b^2 \]Here, \(c\) represents the hypotenuse, the triangle's longest side opposite the right angle, while \(a\) and \(b\) are the other two sides.
To find the length of the hypotenuse, we square both known sides \(a\) and \(b\), then add these values. Finally, we take the square root of this sum to determine \(c\).
This theorem is fundamental in proving that a triangle is right-angled and is heavily used in various mathematical calculations to find missing side lengths. It also lays the groundwork for trigonometric functions that express these side length ratios \(\sin\), \(\cos\), and \(\tan\), ensuring a deep interconnection between geometry and trigonometry.

Sine, Cosine, and Tangent Calculations

Calculating the sine, cosine, and tangent of an angle involves understanding the relative ratios of a right triangle's sides specific to that angle.
Start with the sine of angle A, defined as the opposite side (\(a\)) over the hypotenuse (\(c\)):\[ \sin A = \frac{a}{c} \]For example, if \(a = 85.3\) and \(c = 151.7\), \(\sin A\) becomes approximately 0.56.
Next is cosine, the adjacent side (\(b\)) over the hypotenuse:\[ \cos A = \frac{b}{c} \]Using \(b = 125.5\) and \(c = 151.7\), \(\cos A\) equals about 0.83.
Finally, tangent is the ratio of the opposite side to the adjacent side:\[ \tan A = \frac{a}{b} \]With given values, \(\tan A\) results in around 0.68.
Understanding these calculations allows us to determine not only side lengths but also angles within a right triangle. That's essential for many practical applications like navigation, architecture, and any situations where angle and distance calculations are critical.