Problem 132 - Calculus Volume 1

Question

Consider triangle \(A B C\), 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. \(\quad b=40, c=41\)

Step-by-Step Solution

Answer

The missing side is 9. Trigonometric functions for angle A: \( \sin(A) \approx 1.0 \), \( \cos(A) \approx 0.2 \), \( \tan(A) \approx 4.4 \), \( \csc(A) \approx 1.0 \), \( \sec(A) \approx 4.6 \), and \( \cot(A) \approx 0.2 \).

Step 1: Understand the Problem

We have a right triangle \( ABC \) with a right angle at \( C \). We know \( b = 40 \) and \( c = 41 \). We need to find the length of side \( a \) (hypotenuse) and then calculate the six trigonometric functions for angle \( A \).

Step 2: Use Pythagorean Theorem to Find Missing Side

In a right triangle, the Pythagorean theorem states \( a^2 = b^2 + c^2 \). Here, \( b = 40 \) and the hypotenuse \( c = 41 \). Substitute the values in:\[a^2 = 41^2 - 40^2\a^2 = 1681 - 1600\a^2 = 81\a = \sqrt{81} = 9\].So, the missing side \( a \) is 9.

Step 3: Calculate Sin A

The sine function is defined as the opposite side over hypotenuse. For angle \( A \), \( \sin(A) = \frac{b}{c} = \frac{40}{41} \approx 0.9756 \).

Step 4: Calculate Cos A

The cosine function is defined as the adjacent side over hypotenuse. For angle \( A \), \( \cos(A) = \frac{a}{c} = \frac{9}{41} \approx 0.2195 \).

Step 5: Calculate Tan A

The tangent function is defined as the opposite side over adjacent side. For angle \( A \), \( \tan(A) = \frac{b}{a} = \frac{40}{9} \approx 4.4444 \).

Step 6: Calculate Csc A

The cosecant function is the reciprocal of sine. For angle \( A \), \( \csc(A) = \frac{1}{\sin(A)} \approx \frac{1}{0.9756} \approx 1.025 \).

Step 7: Calculate Sec A

The secant function is the reciprocal of cosine. For angle \( A \), \( \sec(A) = \frac{1}{\cos(A)} \approx \frac{1}{0.2195} \approx 4.556 \).

Step 8: Calculate Cot A

The cotangent function is the reciprocal of tangent. For angle \( A \), \( \cot(A) = \frac{1}{\tan(A)} \approx \frac{1}{4.4444} \approx 0.225 \).

Key Concepts

Right Triangle

In geometry, a right triangle plays a significant role, especially when working with trigonometric functions. A right triangle is a triangle that includes one angle measuring exactly 90 degrees. This 90-degree angle is known as a right angle. In triangle terminology, the sides are classified as follows:

• The side opposite the right angle is called the hypotenuse.
• The two sides forming the right angle are known as the legs.

In the example of triangle ABC that has a right angle at C, and sides b and c are given, the understanding of these side relationships is key to solving for missing lengths and angles. Recognizing which side is opposite and which is adjacent is crucial to using trigonometric functions effectively.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used in right triangles, named after the ancient Greek mathematician Pythagoras. It provides the relationship between the sides of a right triangle, which is expressed as:\[ a^2 = b^2 + c^2 \]where:

• \( a \) represents the hypotenuse,
• \( b \) and \( c \) are the other two sides likewise known as the legs.

In the given problem, with \( b = 40 \) and the hypotenuse \( c = 41 \), the theorem helps us find the missing side \( a \). By substituting the known values into the theorem, we can calculate:\[ a^2 = 41^2 - 40^2 \]\[ a = \sqrt{81} = 9 \]This means the length of side \( a \) is 9.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle and are central to the study of trigonometry. These functions relate the angles of a triangle to the ratios of its sides and include:

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

Each function involves a specific ratio and is extremely useful in calculating unknown sides and angles of triangles. These functions are periodic and repeat their values over specific intervals, which are useful properties when solving problems involving right triangles.

Sine and Cosine

Sine and cosine are perhaps the most well-known trigonometric functions. They are derived from the ratios of the sides of a right triangle.

• The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse: \( \sin(A) = \frac{b}{c} \).
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse: \( \cos(A) = \frac{a}{c} \).

In our scenario, for angle A:

• \( \sin(A) = \frac{40}{41} \approx 0.9756 \)
• \( \cos(A) = \frac{9}{41} \approx 0.2195 \)

Understanding these relationships helps in solving not only geometrical problems but also real-world applications involving angles and distances.

Tangent and Cotangent

The tangent and cotangent functions help further explore the relationships within a right triangle. Tangent is particularly useful in various scientific applications:

• The tangent of an angle is defined as the ratio of the opposite side to the adjacent side: \( \tan(A) = \frac{b}{a} \).
• The cotangent, on the other hand, is the reciprocal of the tangent: \( \cot(A) = \frac{1}{\tan(A)} \).

For triangle ABC, these values are:

• \( \tan(A) = \frac{40}{9} \approx 4.4444 \)
• \( \cot(A) = \frac{1}{4.4444} \approx 0.225 \)

These functions are crucial in understanding the full scope of trigonometry, especially in fields like physics and engineering.

Secant and Cosecant

Secant and cosecant are trigonometric functions that are less commonly used than sine, cosine, or tangent but are still important in certain contexts.

• The secant function is the reciprocal of the cosine function: \( \sec(A) = \frac{1}{\cos(A)} \).
• The cosecant function is the reciprocal of the sine function: \( \csc(A) = \frac{1}{\sin(A)} \).

For angle A in triangle ABC:

• \( \sec(A) = \frac{1}{0.2195} \approx 4.556 \)
• \( \csc(A) = \frac{1}{0.9756} \approx 1.025 \)

These functions often pop up in advanced mathematics and can offer alternate insights into the properties of angles and triangles.