Problem 125 - 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 \(\theta\) $$\cos \theta=\frac{3}{4}$$

Step-by-Step Solution

Answer

The missing side of the triangle is of length \(\sqrt{7}\). The other trigonometric functions of \(\theta\) are: \(\sin{\theta} = \frac{{\sqrt{7}}}{4}\), \(\tan{\theta} = \frac{{\sqrt{7}}}{3}\), \(\csc{\theta} = \frac{4}{{\sqrt{7}}}\), \(\sec{\theta} = \frac{4}{3}\), and \(\cot{\theta} = \frac{3}{{\sqrt{7}}}\).

Step 1: Understand the cosine function and draw the triangle

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. In this case, \(\cos{\theta} = \frac{3}{4}\), which means that the adjacent side is 3 units long and the hypotenuse is 4 units long. Draw a right triangle with these lengths.

Step 2: Calculate the length of the third side

According to the Pythagorean theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, using the Pythagorean theorem, the length of the third side (opposite side) can be calculated by \(\sqrt{4^2 - 3^2} = \sqrt{7}\). This is the missing side of the triangle.

Step 3: Find the values of the other trig functions

Now that have all three sides of the triangle, you can calculate the other trigonometric functions: \(\sin{\theta} = \frac{{opposite side}}{{hypotenuse}} = \frac{{\sqrt{7}}}{4}\) \(\tan{\theta} = \frac{{opposite side}}{{adjacent side}} = \frac{{\sqrt{7}}}{3}\) The reciprocal trig functions are:\(\csc{\theta} = \frac{1}{{\sin{\theta}}} = \frac{4}{{\sqrt{7}}}\)\(\sec{\theta} = \frac{1}{{\cos{\theta}}} = \frac{4}{3}\)\(\cot{\theta} = \frac{1}{{\tan{\theta}}} = \frac{3}{{\sqrt{7}}}\)

Key Concepts

Right Triangle

A right triangle is a kind of triangle that has one angle measuring exactly 90 degrees. This special angle makes calculations straightforward, especially when dealing with trigonometric functions. In right triangles, you will often hear terms like "adjacent," "opposite," and "hypotenuse." These are essential terms to understand:

• Adjacent: The side next to the angle you are working with.
• Opposite: The side directly across from the angle you are examining.
• Hypotenuse: The longest side of the triangle, opposite the right angle.

In our case, the problem states \(\cos \theta = \frac{3}{4}\).This tells us that the length of the adjacent side is 3 units, while the hypotenuse is 4 units.
Sketching this out helps us visualize and solve for the unknowns more easily.
Always remember, creating a visual aid like a triangle sketch is a valuable first step in solving trigonometric problems.

Pythagorean Theorem

The Pythagorean Theorem is a critical principle when dealing with right triangles. It relates the lengths of the sides in a right triangle, providing a foundational mathematical approach to finding unknown side lengths.
The theorem states that:\[a^2 + b^2 = c^2\]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
For our exercise:

• Hypotenuse \(c = 4\)
• Adjacent side \(b = 3\)
• Opposite side \(a = \sqrt{4^2 - 3^2} = \sqrt{7}\)

This calculation shows the square of the hypotenuse equals the sum of the squares of the other two sides, thus confirming our use of the theorem. This result places us closer to finding all necessary trigonometric functions.

Reciprocal Trigonometric Functions

Once you have determined all sides of a right triangle, calculating reciprocal trigonometric functions becomes straightforward. These functions are simply the inverses of the basic functions: sine, cosine, and tangent.

• Sine (\(\sin\)):\ Opposite/Hypotenuse. For this problem: \(\sin \theta = \frac{\sqrt{7}}{4}\).
• Cosecant (\(\csc\)):\ The reciprocal of sine. For \(\sin \theta = \frac{\sqrt{7}}{4}\), \(\csc \theta = \frac{4}{\sqrt{7}}\).
• Cosine (\(\cos\)):\ Already given as \(\frac{3}{4}\).
• Secant (\(\sec\)):\ The reciprocal of cosine. So, \(\sec \theta = \frac{4}{3}\).
• Tangent (\(\tan\)):\ Opposite/Adjacent. Hence, \(\tan \theta = \frac{\sqrt{7}}{3}\).
• Cotangent (\(\cot\)):\ The reciprocal of tangent. Thus, \(\cot \theta = \frac{3}{\sqrt{7}}\).

Understanding these relationships not only solves the problem but also improves your grasp of trigonometry's foundational concepts.