Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\cot \theta=\frac{3}{2}$$

Step-by-Step Solution

Answer

Triangle sides: base=3 units, perpendicular=2 units, hypotenuse=\(\sqrt{13}\) units. Trigonometric functions: \(tan \(\theta\)\) = \(\frac{2}{3}\), \(sin \(\theta\)\) = \(\frac{2}{\sqrt{13}}\), \(cos \(\theta\)\) = \(\frac{3}{\sqrt{13}}\), \(csc \(\theta\)\) = \(\frac{\sqrt{13}}{2}\), \(sec \(\theta\)\) = \(\frac{\sqrt{13}}{3}\).

Step 1: Sketch the Right Triangle

The cotangent of an angle \(\theta\), \(cot \(\theta\)\), is equal to the ratio of the adjacent side (base) to the opposite side (perpendicular) in a right triangle. In this case, \(cot \(\theta\)\) = \(\frac{3}{2}\), hence, the length of the base can be considered as 3 units and the length of the perpendicular can be considered as 2 units. Sketch a right triangle, labeling the base as 3 units, the perpendicular as 2 units and the hypotenuse is yet to be found.

Step 2: Use the Pythagorean Theorem

To find the length of the hypotenuse, use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, it translates into \(c^{2}\) = \(3^{2}\) + \(2^{2}\), where c represents the hypotenuse.

Step 3: Determine the Hypotenuse

Solve the equation \(c^{2}\) = \(3^{2}\) + \(2^{2}\) to find the hypotenuse. This yields \(c^{2}\) = 9 + 4 = 13, and taking the square root of both sides gives c = \(\sqrt{13}\). So, the length of the hypotenuse is \(\sqrt{13}\) units.

Step 4: Calculate the Remaining Trigonometric Functions

With the lengths of all sides of the triangle now known, compute the remaining trigonometric functions as follows: sine \(\theta\) (\(sin \(\theta\)\)) is the ratio of the opposite to the hypotenuse, yielding \(sin \(\theta\)\) = \(\frac{2}{\sqrt{13}}\). Cosine \(\theta\) (\(cos \(\theta\)\)) is the ratio of the adjacent to the hypotenuse, yielding \(cos \(\theta\)\) = \(\frac{3}{\sqrt{13}}\). The tangent of \(\theta\) (\(tan \(\theta\)\)) is the ratio of the opposite to the adjacent, yielding \(tan \(\theta\)\) = \(\frac{2}{3}\). The cosecant of \(\theta\) (\(csc \(\theta\)\)) is the reciprocal of the sine, yielding \(csc \(\theta\)\) = \(\frac{\sqrt{13}}{2}\). The secant of \(\theta\) (\(sec \(\theta\)\)) is the reciprocal of the cosine, yielding \(sec \(\theta\)\) = \(\frac{\sqrt{13}}{3}\).

Key Concepts

The Pythagorean Theorem in Right Triangles

The Pythagorean Theorem is a fundamental principle in geometry that applies to right triangles. This theorem states that for any right triangle, the square of the hypotenuse (the longest side opposite the right angle) equals the sum of the squares of the other two sides (the legs). In formulaic terms, it is expressed as:\[ c^2 = a^2 + b^2 \] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
In our problem, when the base is 3 units and the opposite side is 2 units, the hypotenuse \(c\) can be found by plugging these values into the formula, which leads us to:

• \( c^2 = 3^2 + 2^2 \)
• Solve for \(c\) to find \(c = \sqrt{13}\)

This consistent relationship helps us determine unknown side lengths in right triangles, a crucial step when analyzing angles and trigonometric functions.

Understanding Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are vital in right triangle trigonometry and include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent.
The relationship between these function values is expressed through the sides of the triangle:

• Sine \(\theta \) \((sin \theta)\): \( \frac{\text{Opposite side}}{\text{Hypotenuse}} \)
• Cosine \(\theta \) \((cos \theta)\): \( \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)
• Tangent \(\theta \) \((tan \theta)\): \( \frac{\text{Opposite side}}{\text{Adjacent side}} \)
• Cosecant \(\theta \) \((csc \theta)\): Reciprocal of sine, \(\frac{1}{sin \theta}\)
• Secant \(\theta \) \((sec \theta)\): Reciprocal of cosine, \(\frac{1}{cos \theta}\)
• Cotangent \(\theta \) \((cot \theta)\): Reciprocal of tangent, \(\frac{1}{tan \theta}\)

These functions are used to solve angles and find side lengths, making them essential in trigonometry.

Grasping the Concept of Cotangent

Cotangent is one of the six trigonometric functions and is symbolized as \(cot \theta\). It represents the ratio of the adjacent side to the opposite side in a right triangle.
Mathematically, it is expressed as:\[ cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \]In our exercise, we have \( cot \theta = \frac{3}{2} \), meaning the adjacent side is 3 units and the opposite side is 2 units.
This ratio is crucial for determining other functions like sine, cosine, etc., by allowing one to relate and compare different sides of the triangle based on the given angle \(\theta\). The cotangent, as a reciprocal of the tangent function, provides insight into the angle itself and is used to find angles when side lengths are known, or to solve geometric problems involving right triangles, making it a valuable tool in trigonometry.