Problem 14 - Precalculus: A Concise Course

Question

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\boldsymbol{\theta}\). Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \cos \theta=\frac{5}{6} $$

Step-by-Step Solution

Answer

After solving, the opposite side of the triangle to theta is found to be \(\sqrt{11}\). This leads to calculated values of the other trigonometric functions as follows: \(\sin \theta = \frac{\sqrt{11}}{6}\), \(\tan \theta = \frac{\sqrt{11}}{5}\), \(\csc \theta = \frac{6}{\sqrt{11}}\), \(\sec \theta = \frac{6}{5}\), and \(\cot \theta = \frac{5}{\sqrt{11}}\)

Step 1: Drawing the Right Triangle

Considering the definition of cosine that is, \(\cos \theta = \frac{Adjacent\ Side}{Hypotenuse}\), sketch a right triangle. Label one of the non-right angles as \(\ theta \). Label the side of the triangle adjacent to \(\ theta \) as 5 (from \(\ cos \ theta = \frac{5}{6}\)) and the hypotenuse as 6.

Step 2: Calculating the Opposite Side

Using the Pythagorean theorem, \(a^2 + b^2 = c^2 \), where \(a\) and \(b\) represent the two sides of the triangle and \(c\) is the hypotenuse, calculate the length of the remaining side of the triangle (opposite side). Plugging in our known values: \((opposite\ side)^2 + 5^2 = 6^2\). Solve this to find the length of the opposite side.

Step 3: Finding other Trigonometric Functions values

Use the lengths of the sides to calculate the values of the remaining trigonometric functions: \(\sin \theta = \frac{Opposite\ Side}{Hypotenuse}\), \(\tan \theta = \frac{Opposite\ Side}{Adjacent\ Side}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Plug the known side lengths into these formulas to find the corresponding trigonometric values.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are essential when studying triangles, especially right triangles. In a right triangle, we have three main trigonometric functions based on an acute angle, such as \( \theta \). These functions relate the angles to the ratios of the sides of the triangle, making them extremely useful in various mathematical calculations and real-world applications.

• Sine (\( \sin \theta \)): This function is defined as the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): This is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): This is the ratio of the opposite side to the adjacent side.

Apart from these, we also have the reciprocal functions:

• Cosecant (\( \csc \theta \)), the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \)
• Secant (\( \sec \theta \)), the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cotangent (\( \cot \theta \)), the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \)

These functions help us find missing sides and angles in triangles and are fundamental in calculus and beyond.

Exploring the Pythagorean Theorem

The Pythagorean Theorem is a cornerstone in geometry that describes the relationship between the sides of a right triangle. It defines how the hypotenuse (the side opposite the right angle) is related to the other two sides. This theorem is given by the formula: \[a^2 + b^2 = c^2\]where \( a \) and \( b \) are the lengths of the two shorter sides, and \( c \) is the length of the hypotenuse.
To apply the theorem, consider our example where \( \cos \theta = \frac{5}{6} \). We label the adjacent side \( a = 5 \) and the hypotenuse \( c = 6 \). Using the theorem, we can calculate the opposite side \( b \) as follows:\[b^2 + 5^2 = 6^2\]Solve for \( b \) to find the length of the opposite side, completing the triangle. Knowing all three sides allows you to calculate all the trigonometric functions of angle \( \theta \).

Diving into the Cosine Function

The cosine function is frequently used in various fields, from engineering to physics. It helps in determining the horizontal component of a vector, among other applications. In the context of a right triangle, \( \cos \theta \) is defined very simply as:\[\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\]In our problem, we have \( \cos \theta = \frac{5}{6} \), indicating that for the angle \( \theta \), the adjacent side measures 5, and the hypotenuse measures 6.
This ratio provides much more than just a triangle property; it offers insight into the angle's nature. It can be extended to the unit circle where cosine values are crucial for defining coordinates. By knowing \( \cos \theta \), you can quickly compute \( \sec \theta \) because it is simply the reciprocal, \( \sec \theta = \frac{1}{\cos \theta} \), which in this case would be \( \frac{6}{5} \).
Understanding these fundamentals of \( \cos \theta \) enables deeper insights into more complex trigonometric concepts and their applications.