Problem 21 - Algebra 2

Question

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((-\sqrt{3},-\sqrt{6})\)

Step-by-Step Solution

Answer

The six trigonometric functions are: \(\sin \theta = -\frac{\sqrt{6}}{3}\), \(\cos \theta = -\frac{\sqrt{3}}{3}\), \(\tan \theta = \sqrt{2}\), \(\csc \theta = -\frac{\sqrt{6}}{2}\), \(\sec \theta = -\sqrt{3}\), and \(\cot \theta = \frac{\sqrt{2}}{2}\).

Step 1: Calculate the Radius or Hypotenuse

To find the exact values of the trigonometric functions, we first calculate the radius (often called the hypotenuse in a right triangle) using the Pythagorean theorem. The radius is given by:\[ r = \sqrt{x^2 + y^2} \]Substituting the given values, \(x = -\sqrt{3}\) and \(y = -\sqrt{6}\), we get:\[ r = \sqrt{(-\sqrt{3})^2 + (-\sqrt{6})^2} = \sqrt{3 + 6} = \sqrt{9} = 3 \]

Step 2: Find the Sine of \(\theta\)

The sine function is defined as the ratio of the opposite side to the hypotenuse in a right triangle. Thus, \(\sin \theta\) is given by:\[ \sin \theta = \frac{y}{r} = \frac{-\sqrt{6}}{3} \]

Step 3: Find the Cosine of \(\theta\)

The cosine function is the ratio of the adjacent side to the hypotenuse. Hence, \(\cos \theta\) is:\[ \cos \theta = \frac{x}{r} = \frac{-\sqrt{3}}{3} \]

Step 4: Find the Tangent of \(\theta\)

The tangent function is the ratio of the opposite side to the adjacent side. Using the given values:\[ \tan \theta = \frac{y}{x} = \frac{-\sqrt{6}}{-\sqrt{3}} = \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{2} \]

Step 5: Find the Cosecant of \(\theta\)

The cosecant function is the reciprocal of the sine function. Thus, \(\csc \theta\) is:\[ \csc \theta = \frac{1}{\sin \theta} = \frac{3}{-\sqrt{6}} = -\frac{\sqrt{6}}{2} \]

Step 6: Find the Secant of \(\theta\)

The secant function is the reciprocal of the cosine function. Hence, \(\sec \theta\) is:\[ \sec \theta = \frac{1}{\cos \theta} = \frac{3}{-\sqrt{3}} = -\sqrt{3} \]

Step 7: Find the Cotangent of \(\theta\)

The cotangent function is the reciprocal of the tangent function. Therefore, \(\cot \theta\) is:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental tool in trigonometry, especially useful when working with right triangles. It states that, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is expressed in the formula:

• \(c^2 = a^2 + b^2\)

In the context of trigonometry, we often refer to the hypotenuse as \(r\), and the two other sides as \(x\) and \(y\), which typically denote the lengths on the coordinate axes.

For the problem at hand, the given point \((-\sqrt{3}, -\sqrt{6})\) means \(x = -\sqrt{3}\) and \(y = -\sqrt{6}\). We utilize the Pythagorean theorem to find \(r\) (radius or hypotenuse):

• \(r = \sqrt{x^2 + y^2} = \sqrt{(-\sqrt{3})^2 + (-\sqrt{6})^2} = 3\)

This gives us the length of the radius, which is crucial for calculating the trigonometric functions.

sine function

The sine function in trigonometry relates to the angle of a right triangle and is defined as the ratio of the length of the opposite side to the hypotenuse. Its formula is:

• \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)

For the given problem, after finding \(r = 3\), the sine of \(\theta\) is calculated using \(y\):

• \(\sin \theta = \frac{y}{r} = \frac{-\sqrt{6}}{3}\)

This value is the sine of the angle in our coordinate system. Since both components are negative, the terminal side is in the third quadrant of the unit circle.

cosine function

The cosine function, another fundamental trigonometric ratio, represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle. It's calculated using the formula:

• \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)

In our example, with the hypotenuse \(r\) established at 3, we find:

• \(\cos \theta = \frac{x}{r} = \frac{-\sqrt{3}}{3}\)

This value shows the cosine of \(\theta\), indicating the horizontal distance relative to the hypotenuse. The negative sign reveals it's pointing in the opposite direction along the x-axis.

tangent function

The tangent function is a trigonometric function that delineates the ratio of the opposite side to the adjacent side of a right triangle. Mathematically, it is expressed as:

• \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)

To calculate this for \(\theta\), based on our given values, we use:

• \(\tan \theta = \frac{y}{x} = \frac{-\sqrt{6}}{-\sqrt{3}} = \sqrt{2}\)

This result for tangent indicates the steepness of the angle \(\theta\) in our coordinate plane and illustrates how the y-component influences the angle's direction.

reciprocal functions

In trigonometry, reciprocal functions extend the basic ratios to provide more flexibility in calculations and interpretations. These are:

• Cosecant \(\csc \theta = \frac{1}{\sin \theta}\)
• Secant \(\sec \theta = \frac{1}{\cos \theta}\)
• Cotangent \(\cot \theta = \frac{1}{\tan \theta}\)

For our example:

• \(\csc \theta = \frac{1}{\sin \theta} = -\frac{\sqrt{6}}{2}\)
• \(\sec \theta = \frac{1}{\cos \theta} = -\sqrt{3}\)
• \(\cot \theta = \frac{1}{\tan \theta} = \frac{\sqrt{2}}{2}\)

These functions offer insight into the reciprocal relationships within right triangles and highlight different aspects of angles and their measures in trigonometry.