Problem 14 - Precalculus

Question

The terminal side of an angle \(\theta\) in standard position passes through values of the six trigonometric functions for angle \(\theta\) $$\left(-\frac{2}{9},-\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

\(\sin(\theta) = -\frac{3\sqrt{13}}{13}, \cos(\theta) = -\frac{2\sqrt{13}}{13}, \tan(\theta) = \frac{3}{2}, \csc(\theta) = -\frac{\sqrt{13}}{3}, \sec(\theta) = -\frac{\sqrt{13}}{2}, \cot(\theta) = \frac{2}{3}.\)

Step 1: Find the distance from the origin

The point \( \left(-\frac{2}{9}, -\frac{1}{3}\right) \) must lie on the terminal side of angle \( \theta \) in standard position. To find the length of the radius (r) in the unit circle through this point, use the Pythagorean theorem: \[ r = \sqrt{\left(-\frac{2}{9}\right)^2 + \left(-\frac{1}{3}\right)^2} \]First, square both components:\[ \left(-\frac{2}{9}\right)^2 = \frac{4}{81} \]\[ \left(-\frac{1}{3}\right)^2 = \frac{1}{9} = \frac{9}{81} \]Adding these gives:\[ r = \sqrt{\frac{4}{81} + \frac{9}{81}} = \sqrt{\frac{13}{81}} = \frac{\sqrt{13}}{9} \]So, the radius \( r = \frac{\sqrt{13}}{9} \).

Step 2: Calculate the sine and cosine of \(\theta\)

The sine and cosine are given by the coordinates of the point divided by the radius:\[ \sin(\theta) = \frac{y}{r} = \frac{-\frac{1}{3}}{\frac{\sqrt{13}}{9}} = -\frac{9}{3\sqrt{13}} = -\frac{3}{\sqrt{13}} \]To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{13} \):\[ \sin(\theta) = -\frac{3\sqrt{13}}{13} \]Similarly, for cosine:\[ \cos(\theta) = \frac{x}{r} = \frac{-\frac{2}{9}}{\frac{\sqrt{13}}{9}} = -\frac{2}{\sqrt{13}} \]Rationalize the denominator:\[ \cos(\theta) = -\frac{2\sqrt{13}}{13} \]

Step 3: Define the tangent of \(\theta\)

The tangent function is the ratio of the sine to the cosine:\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{3\sqrt{13}}{13}}{-\frac{2\sqrt{13}}{13}} = \frac{-3\sqrt{13} \cdot 13}{13 \cdot -2\sqrt{13}} = \frac{3}{2} \]

Step 4: Calculate the other trigonometric functions

For the other trigonometric functions, use the reciprocal identities:1. Cosecant (csc) is the reciprocal of sine:\[ \csc(\theta) = \frac{1}{\sin(\theta)} = -\frac{\sqrt{13}}{3} \]2. Secant (sec) is the reciprocal of cosine:\[ \sec(\theta) = \frac{1}{\cos(\theta)} = -\frac{\sqrt{13}}{2} \]3. Cotangent (cot) is the reciprocal of tangent:\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{2}{3} \]

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding relationships within a triangle. They relate angles to side lengths and include six main functions: sine, cosine, tangent, cosecant, secant, and cotangent.

• Sine (\( ext{sin} \theta \)): Measures the ratio of the opposite side to the hypotenuse.
• Cosine (\( ext{cos} \theta \)): The ratio of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan} \theta \)): Defined as the ratio between the opposite and adjacent sides.
• Cosecant (\( ext{csc} \theta \)): Reciprocal of sine.
• Secant (\( ext{sec} \theta \)): Reciprocal of cosine.
• Cotangent (\( ext{cot} \theta \)): Reciprocal of tangent.

In problems involving points not on the circle, using these functions becomes crucial, as seen in the given exercise. Knowing how to find these functions helps simplify complex calculations in various mathematical and engineering problems.

Unit Circle

The unit circle is a foundational concept in trigonometry. It is a circle with a radius of one unit and centered at the origin \( (0, 0) \) in the coordinate plane. This circle enables easy calculation of trigonometric functions for various angles.
Key aspects to consider when using the unit circle:

• The radius is always 1, aiding calculations and visualizations.
• Angles are measured from the positive x-axis, moving counterclockwise for positive angles.
• Coordinates on the unit circle correspond to cosine and sine values for a given angle.

In the exercise, although the point is not on the unit circle, we use the concept to understand how \( heta \) relates to known trigonometric functions by calculating distances and dividing by the radius. This approach extends the utility of the unit circle beyond simple textbook cases.

Pythagorean Theorem

The Pythagorean theorem is a critical concept in understanding the relationships between the sides of a right triangle. It is expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
The theorem is instrumental in trigonometry for calculating distances, such as when dealing with the terminal side of angles in exercises. Wherever there is a point not lying on an axis, like \( ext{(-}\frac{2}{9}\text{,-}\frac{1}{3}) \), this theorem helps find its Euclidean distance from the origin (also known as the radius in unit circle terminology).

• This distance informs the sine and cosine values in trig computations as seen.
• Despite the complex-looking coordinates, the Pythagorean theorem easily navigates these complexities to yield straightforward results.

Among its numerous applications, it is a simple yet powerful tool to bridge the gap between algebra and geometry through trigonometry.