Problem 126 - 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 and then find the values of the other five trigonometric functions of \(\theta\) \(\sec \theta=3\)

Step-by-Step Solution

Answer

The other trigonometric functions are: \(\sin \theta = \frac{\sqrt{8}}{3}\), \(\tan \theta = \sqrt{8}\), \(\csc \theta = \frac{3}{\sqrt{8}}\), \(\cot \theta = \frac{1}{\sqrt{8}}\).

Step 1: Identifying the adjacent side and the hypotenuse from \(\cos \theta\)

Given that \(\cos \(\theta\) = 1/3\), and cos is the ratio of the adjacent side (a) to the hypotenuse (h) in a right-triangle, we can let the adjacent side to be 1 (a = 1) and the hypotenuse to be 3 (h = 3).

Step 2: Calculating the opposite side using the Pythagorean theorem

The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the lengths of the other two sides (a and b). Therefore, \(b = \sqrt{h^2 - a^2} = \sqrt{3^2 - 1^2} = \sqrt{8}\).

Step 3: Finding the other trigonometric functions

Now that we have all the sides of the triangle and the angle \(\theta\), we can calculate the other trigonometric functions: \(\sin \theta = \frac{o}{h} = \frac{\sqrt{8}}{3}\), \(\tan \theta = \frac{o}{a} = \sqrt{8}\), \(\csc \theta = \frac{1}{\sin \theta} = \frac{3}{\sqrt{8}}\), \(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{8}}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This right angle distinguishes it from other triangles and plays a key role in trigonometry. In a right triangle, the longest side is called the hypotenuse, which is opposite the right angle. The other two sides are known as the legs. These sides form the right angle and are referred to as the adjacent and opposite sides, depending on their relation to a given angle within the triangle.
In the context of trigonometric functions, the particular triangle often serves as a basis to define various ratios such as sine, cosine, and tangent, which relate the sides to an acute angle in the triangle.
In our exercise, the given trigonometric information helps us to visually and mathematically understand how these sides are positioned and calculated.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies specifically to right triangles. It states that 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 ( a and b). The formula is expressed as: \( a^2 + b^2 = c^2 \) .
This theorem provides a reliable method to calculate the unknown length of a triangle's side when the other two are known.
In the exercise, we used the theorem to find the opposite side of the triangle when given the hypotenuse and adjacent side. By substituting the known values into the Pythagorean formula, we determined the length of the missing side as \( b = \sqrt{8} \) . This calculation is essential to establish a complete understanding of the triangle’s dimensions.

Acute Angle

An acute angle is an angle that measures less than 90 degrees. These angles are common components of right triangles, as one of their angles is always a right angle, the remaining two must be acute.
In the context of trigonometry and our exercise, we focus on one of these acute angles, \( \theta \) , which is essential for calculating the trigonometric functions.
Understanding which side is the adjacent versus the opposite in relation to the acute angle is crucial because it dictates how we apply functions like sine, cosine, and tangent. In the given triangle scenario, \( \theta \) is the focus, and from its perspective, we identify the sides correctly to use trigonometric relationships.

Secant Function

The secant function ( \( \sec \theta \) ) is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. If \( \cos \theta \) is the ratio of the adjacent side to the hypotenuse, then \( \sec \theta \) is the inverse, defined as the hypotenuse divided by the adjacent side: \( \sec \theta = \frac{h}{a} \).
In practical terms, in a right triangle context, it tells us the factor by which the hypotenuse is stretched relative to the adjacent side when the angle \( \theta \) is considered.
In the problem, \( \sec \theta \) is provided as a starting point, which equals 3. This information allowed us to determine both the hypotenuse and adjacent side of the triangle, leading to the identification of the remaining trigonometric functions.