Problem 27 - Algebra 2

Question

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\cos \theta=\frac{1}{5}\)

Step-by-Step Solution

Answer

The six trigonometric ratios of the angle \(\theta\) are: \(\cos \theta = \frac{1}{5}, \sin \theta = \frac{\sqrt{24}}{5}, \tan \theta = \sqrt{24}, \csc \theta = \frac{5}{\sqrt{24}}, \sec \theta = 5, \cot \theta = \frac{1}{\sqrt{24}}\).

Step 1: Sketch the right triangle with the given information

Draw a right triangle and let one of the acute angles be \(\theta\). By definition, \(\cos \theta\) is the ratio of the adjacent side to the hypotenuse. Since \(\cos \theta=\frac{1}{5}\), label the adjacent side 1 and the hypotenuse 5.

Step 2: Find the length of the opposite side

The opposite side can be found by using 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. Mathematically, this can be expressed as \(c^2 = a^2 + b^2\), where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Rearranging the formula to solve for the opposite side we will get \(a = \sqrt{c^2 - b^2}\). Substituting the given values, \(a = \sqrt{5^2 - 1^2} = \sqrt{24}\). So, the length of the opposite side is \(\sqrt{24}\)

Step 3: Compute the other trigonometric ratios

Now, we can use the lengths of the sides to compute the other trigonometric ratios.1. \(\sin \theta\) is the ratio of the opposite side to the hypotenuse. Hence, \(\sin \theta= \frac{\sqrt{24}}{5}\)2. \(\tan \theta\) is the ratio of the opposite side to the adjacent side. Hence, \(\tan \theta= \sqrt{24}\)3. \(\csc \theta\) is the reciprocal of \(\sin \theta\). Hence, \(\csc \theta= \frac{5}{\sqrt{24}}\)4. \(\sec \theta\) is the reciprocal of \(\cos \theta\). Hence, \(\sec \theta= 5\)5. \(\cot \theta\) is the reciprocal of \(\tan \theta\). Hence, \(\cot \theta= \frac{1}{\sqrt{24}}\)

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This distinguishing feature allows us to use various mathematical properties that are unique to right triangles. In a right triangle:

• The side opposite the right angle is known as the hypotenuse, and it is the longest side.
• The other two sides are called the legs of the triangle.
• The angles that are not the right angle are referred to as acute angles, each being less than 90 degrees.

In this exercise, when the angle \(\theta\) is specified as one of the acute angles, we're working within the framework of a right triangle, drawing on its special properties to solve for trigonometric ratios.

Pythagorean Theorem

The Pythagorean theorem is a powerful tool in geometry, especially useful in right triangles. It states that the square of the hypotenuse's length (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The mathematical expression of this theorem is:\[c^2 = a^2 + b^2\]Where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.
This formula comes in handy when you know the length of two sides of a right triangle and need to find the third. In our exercise, we used this theorem to find the length of the opposite side given one leg and the hypotenuse. By plugging in the known values, we found:\[a = \sqrt{5^2 - 1^2} = \sqrt{24}\]This result helps us determine the other trigonometric ratios.

Acute Angle

An acute angle is any angle less than 90 degrees. In the context of a right triangle, the two angles besides the right angle are acute angles.
These angles are crucial in defining trigonometric ratios like sine, cosine, and tangent, as they are based on the relationships of the triangle's side lengths relative to these acute angles.
Knowing \(\cos \theta = \frac{1}{5}\) tells us that \(\theta\) is one of these acute angles, and it starts the process for defining all six primary trigonometric ratios by using the lengths of the sides that correspond to \(\theta\).

• The cosine of \(\theta\) involves the adjacent side and hypotenuse.
• The sine of \(\theta\) involves the opposite side and hypotenuse.
• The tangent of \(\theta\) involves the opposite and adjacent sides.

Understanding the role of acute angles helps navigate problems involving trigonometry effectively.

Reciprocal Identities

Reciprocal identities are relations between trigonometric functions derived from their basic definitions. They link primary trigonometric functions to their reciprocals and are useful when solving problems or finding unknown ratios:

• \(\csc \theta\) is the reciprocal of \(\sin \theta\), calculated by flipping the ratio of sine, thus: \(\csc \theta = \frac{1}{\sin \theta}\).
• \(\sec \theta\) is the reciprocal of \(\cos \theta\), so \(\sec \theta = \frac{1}{\cos \theta}\).
• \(\cot \theta\) is the reciprocal of \(\tan \theta\), so \(\cot \theta = \frac{1}{\tan \theta}\).

In our exercise, these identities allow us to find the remaining trigonometric ratios without additional computation. For example, the reciprocal of \(\cos \theta = \frac{1}{5}\) immediately gives us \(\sec \theta = 5\). Such identities are fundamental when delving deeper into trigonometric functions and their applications.