Problem 37 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). \(\left(-\frac{20}{29}, \frac{21}{29}\right)\)

Step-by-Step Solution

Answer

\(\sin t = \frac{21}{29}, \cos t = -\frac{20}{29}, \tan t = -\frac{21}{20}.\)

Step 1: Identify the coordinates of P

The terminal point \( P(x, y) \) is given as \( \left(-\frac{20}{29}, \frac{21}{29}\right) \). Here, \( x = -\frac{20}{29} \) and \( y = \frac{21}{29} \).

Step 2: Calculate \(\sin t\)

Since the terminal point is on the unit circle, \( \sin t \) is given by the \( y \)-coordinate of \( P \). Thus, \[ \sin t = \frac{21}{29}. \]

Step 3: Calculate \(\cos t\)

Since the terminal point is on the unit circle, \( \cos t \) is given by the \( x \)-coordinate of \( P \). Thus, \[ \cos t = -\frac{20}{29}. \]

Step 4: Calculate \(\tan t\)

\( \tan t \) is given by the ratio \( \frac{y}{x} \). Thus, \[ \tan t = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20}. \]

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, used to define trigonometric functions for all real numbers. It is a circle with a radius of 1 centered at the origin in the coordinate plane. Each point on the unit circle corresponds to an angle measured from the positive x-axis, and the coordinates of the point provide the values of the sine and cosine functions for that angle. Since the radius is 1, the equation of the unit circle is given by: \[ x^2 + y^2 = 1. \] This equation ensures that for any angle, the sum of the squares of the sine and cosine of the angle equals 1, reflecting the Pythagorean identity. By using the unit circle, we can easily visualize how angle measures relate to the coordinates \( (x, y) \) of the points on the circle.

Sine Function

The sine function, often abbreviated as \( \sin \), is one of the primary trigonometric functions. For a given angle \( t \), \( \sin t \) represents the y-coordinate of the corresponding point on the unit circle. This function is periodic with a period of \( 2\pi \), meaning that its values repeat every \( 2\pi \) units.

• Range: The values of \( \sin t \) lie between -1 and 1.
• Maximum and Minimum: The maximum value is 1, and the minimum value is -1.
• Zeros: \( \sin t \) is zero at integer multiples of \( \pi \).

When given a point \( P(x, y) \) on the unit circle, such as \( \left(-\frac{20}{29}, \frac{21}{29}\right) \), the sine of the angle \( t \) is the y-coordinate, which is \( \frac{21}{29} \). Understanding this relationship can help you comprehend how sine behaves in different quadrants of the circle.

Cosine Function

The cosine function, denoted \( \cos \), is another essential trigonometric function. Like sine, it is defined using the unit circle and for an angle \( t \), \( \cos t \) represents the x-coordinate of the point on the unit circle.

• Range: The range of \( \cos t \) is also between -1 and 1.
• Maximum and Minimum: It reaches its maximum value of 1 at even multiples of \( \pi \) radians and its minimum value of -1 at odd multiples of \( \pi \).
• Zeros: The function is zero at odd multiples of \( \frac{\pi}{2} \).

For the point \( \left(-\frac{20}{29}, \frac{21}{29}\right) \), cos(t) is the x-coordinate, which results in \( -\frac{20}{29} \). The properties of \( \cos \) function help us determine its exact behavior and relationship to the unit circle.

Tangent Function

The tangent function, abbreviated as \( \tan \), represents the ratio of the sine of an angle to the cosine of that angle, expressed as \( \tan t = \frac{\sin t}{\cos t} \). On the unit circle, this means dividing the y-coordinate by the x-coordinate to get the tangent value.

• Periodicity: The tangent function repeats every \( \pi \) units, unlike sine and cosine that repeat every \( 2\pi \) units.
• Undefined Points: It is undefined at angles where \( \cos t = 0 \), as division by zero is not possible.
• Zeros: The tangent is zero at integer multiples of \( \pi \).

For the given point \( \left(-\frac{20}{29}, \frac{21}{29}\right) \) on the unit circle, \( \tan t \) is calculated as the ratio \( \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20} \). This result shows how important it is to recognize the signs and values in different quadrants.