Problem 32 - Precalculus Mathematics for Calculus

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(-\frac{1}{3},-\frac{2 \sqrt{2}}{3}\right)$$

Step-by-Step Solution

Answer

\( \sin t = -\frac{2\sqrt{2}}{3}, \cos t = -\frac{1}{3}, \tan t = 2\sqrt{2} \).

Step 1: Identify Values

We are given the terminal point \( P(x, y) = \left(-\frac{1}{3}, -\frac{2\sqrt{2}}{3}\right) \). This means \( x = -\frac{1}{3} \) and \( y = -\frac{2\sqrt{2}}{3} \).

Step 2: Calculate Radius (r)

The radius \( r \) can be calculated using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the values for \( x \) and \( y \):\[r = \sqrt{\left(-\frac{1}{3}\right)^2 + \left(-\frac{2\sqrt{2}}{3}\right)^2} = \sqrt{\frac{1}{9} + \frac{8}{9}} = \sqrt{1} = 1\]

Step 3: Find \( \\cos t \)

Using the definition \( \cos t = \frac{x}{r} \), substitute \( x = -\frac{1}{3} \) and \( r = 1 \): \[\cos t = \frac{-\frac{1}{3}}{1} = -\frac{1}{3}\]

Step 4: Find \( \\sin t \)

Similarly, use the definition \( \sin t = \frac{y}{r} \). Substitute \( y = -\frac{2\sqrt{2}}{3} \) and \( r = 1 \): \[\sin t = \frac{-\frac{2\sqrt{2}}{3}}{1} = -\frac{2\sqrt{2}}{3}\]

Step 5: Find \( \\tan t \)

The tangent is given by \( \tan t = \frac{y}{x} \). Substitute the values \( x = -\frac{1}{3} \) and \( y = -\frac{2\sqrt{2}}{3} \): \[\tan t = \frac{-\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = 2\sqrt{2}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations in trigonometry that hold true for all values of the involved variables. These identities serve as tools to simplify expressions and solve trigonometric equations. Understanding these can make working with trigonometric functions easier and more intuitive.

Some commonly used trigonometric identities include:

• Pythagorean identity: This is expressed as \( \sin^2\theta + \cos^2\theta = 1 \). It shows a relationship between sine and cosine.
• Reciprocal identities: These are \( \sin\theta = \frac{1}{\csc\theta} \) and \( \cos\theta = \frac{1}{\sec\theta} \), among others.
• Angle sum identities: These involve adding angles, such as \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \).

By utilizing these identities, computations that initially seem complex can often be simplified. This is particularly helpful when solving problems involving trigonometric expressions and equations.

Trigonometric Ratios

Trigonometric ratios are another core concept, providing a link between the angles of a triangle and the ratios of its sides. These ratios are foundational in understanding how different angles relate to each other geometrically.

There are three primary trigonometric functions known as ratios:

• Sine (): It is defined as \( \sin\theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \).
• Cosine (): It is defined as \( \cos\theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).
• Tangent (): It is defined as \( \tan\theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \).

These ratios are profoundly useful when studying right-angled triangles, as they allow for easy calculation of unknown sides or angles in a triangle. Furthermore, these functions extend beyond right triangles and are used to solve real-world problems involving wave patterns, sound, and circular movement.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is a crucial tool in trigonometry and helps to define trigonometric functions for all real numbers.

On the unit circle:

• The x-coordinate of a point on the circle is the cosine of the angle.
• The y-coordinate of a point is the sine of the angle.
• The angle in radians measures the arc length from the positive x-axis to the point.

By using the unit circle, you can easily find the sine and cosine values corresponding to any angle, which then enable calculations for tangent and other trigonometric functions. For example, any point \( P(x, y) \) on the unit circle can be expressed as \( P(\cos t, \sin t) \). This representation makes it simpler to visualize how trigonometric functions behave as the angle changes and emphasizes the cyclic nature of these functions, repeating every \( 2\pi \).

The unit circle thus becomes a visual blueprint to understand circular functions and their properties across different quadrants.