Question

Point \(\mathrm{P}(x, y)\) is plotted where the terminal arm of angle \(\theta\) intersects the unit circle. a) Use \(\mathrm{P}(x, y)\) to determine the slope of the terminal arm. b) Explain how your result from part a) is related to tan \(\theta\) c) Write your results for the slope from part a) in terms of sine and cosine. d) From your answer in part c), explain how you could determine tan \(\theta\) when the coordinates of point \(P\) are known.

Short answer

The slope is \( \frac{y}{x} \), which is \( \tan(\theta) \) and can be expressed as \( \frac{\sin(\theta)}{\cos(\theta)} \).

Step-by-step solution

Identify Point P

Point P has coordinates (x, y) and lies on the unit circle where the terminal arm of angle \( \theta \) intersects.

Determine the Slope of the Terminal Arm

The slope of a line through the origin (0, 0) and point P(x, y) is given by the rise over run: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{y - 0}{x - 0} = \frac{y}{x}. \] Thus, the slope of the terminal arm is \( \frac{y}{x} \).

Relate the Slope to \( \tan(\theta) \)

On the unit circle, \( \tan(\theta) \) is defined as the ratio of the y-coordinate to the x-coordinate: \[ \tan(\theta) = \frac{y}{x}. \] Therefore, the slope of the terminal arm \( \frac{y}{x} \) is equal to \( \tan(\theta) \).

Express the Slope in Terms of Sine and Cosine

On the unit circle, the coordinates (x, y) are given by \( (\cos(\theta), \sin(\theta)) \). Thus, \[ \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}. \] So, the slope in terms of sine and cosine is \( \frac{\sin(\theta)}{\cos(\theta)}. \)

Determine \( \tan(\theta) \) Using Coordinates

From part c), it is clear that \( \tan(\theta) \) is given by \( \frac{\sin(\theta)}{\cos(\theta)} \). Therefore, when the coordinates of point P are known, \( \tan(\theta) \) can be calculated as \[ \tan(\theta) = \frac{y}{x}. \]

Key concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of 1 unit centered at the origin (0, 0) in the Cartesian coordinate system. Any point on the unit circle corresponds to an angle measured from the positive x-axis. Each point can be represented as \( (\cos(\theta), \sin(\theta)) \), where \( \theta \) is the angle in radians.

Understanding points on the unit circle is crucial because it links geometric concepts to trigonometric functions.

• Every point \( P(x, y) \) on the unit circle satisfies the equation \( x^2 + y^2 = 1 \).
• Each coordinate represents cosine and sine values, respectively: \( x = \cos(\theta) \) and \( y = \sin(\theta) \).

By plotting various points, the unit circle helps visualize how sine and cosine values change with the angle \( \theta \). This visual understanding is useful for solving trigonometric problems and proving identities.

Sine and Cosine

Sine and cosine are foundational trigonometric functions derived from the coordinates on the unit circle. They are used to describe the relationship between an angle and the ratios of sides in a right triangle.

\( \cos(\theta) \) is the x-coordinate and \( \sin(\theta) \) is the y-coordinate for any angle \( \theta \) on the unit circle.

• \( \cos(\theta) = \frac{Adjacent}{Hypotenuse} \)
• \( \sin(\theta) = \frac{Opposite}{Hypotenuse} \)

These functions oscillate between -1 and 1 as \( \theta \) varies from 0 to \( 2\pi \) radians. Here are some key points:

• When \( \theta = 0 \), \( \cos(0) = 1 \) and \( \sin(0) = 0 \).
• At \( \frac{\pi}{2} \), \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \).
• When \( \theta = \pi \), \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \).
• Finally, at \( \frac{3\pi}{2} \), \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \).

Tangent

Tangent is another essential trigonometric function. It is related to both sine and cosine and represents the slope of the terminal arm in the unit circle. Mathematically, it is defined as the ratio of the sine and cosine of an angle:

\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
This relationship holds a few significant properties:

• Tangent is undefined for angles where \( \cos(\theta) = 0 \), specifically at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).
• Tangent function has a periodicity of \( \pi \), meaning \( \tan(\theta + \pi) = \tan(\theta) \).

This function extends from negative infinity to positive infinity, making it useful for capturing a wide range of values. In the context of a right triangle, tangent can also be expressed as:

\[ \tan(\theta) = \frac{Opposite}{Adjacent} \]
This definition is identical to the slope formula \( \frac{y}{x} \) mentioned earlier. Thus, \( \tan(\theta) \) efficiently links angles to slopes and is widely used in various trigonometric applications.