Problem 29 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). \(\left(\frac{3}{5}, \frac{4}{5}\right)\)

Step-by-Step Solution

Answer

\(\sin t = \frac{4}{5}, \cos t = \frac{3}{5}, \tan t = \frac{4}{3}\).

Step 1: Analyze the coordinates

The terminal point given is \(P\left(x, y\right) = \left(\frac{3}{5}, \frac{4}{5}\right)\). This point corresponds to a point on the unit circle because the x-coordinate \(x = \frac{3}{5}\) and y-coordinate \(y = \frac{4}{5}\) satisfy the equation \(x^2 + y^2 = 1\) as we'll verify.

Step 2: Verify the point on the unit circle

Calculate \(x^2 + y^2\) to ensure it equals 1. \[ \left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 = \frac{9}{25} + \frac{16}{25} = \frac{25}{25} = 1 \] The point is indeed on the unit circle.

Step 3: Determine \(\sin t\)

On the unit circle, the y-coordinate of the terminal point represents \(\sin t\). Thus, \(\sin t = \frac{4}{5}\).

Step 4: Determine \(\cos t\)

Similarly, the x-coordinate of the terminal point represents \(\cos t\). Therefore, \(\cos t = \frac{3}{5}\).

Step 5: Determine \(\tan t\)

The tangent of angle \(t\) is given by the ratio of the sine and cosine: \(\tan t = \frac{\sin t}{\cos t} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit centered at the origin of a coordinate plane. Every point on this circle satisfies the equation \(x^2 + y^2 = 1\). This simple circle serves as an intuitive tool to understand trigonometric functions based on angles and lengths. The real beauty of the unit circle lies in its ability to connect geometric angles with trigonometric functions.
On the unit circle, an angle \(t\) measured in radians from the positive x-axis correlates directly to a unique point \(P(x, y)\) on the circle. These coordinates \(x\) and \(y\) can then be used to determine the primary trigonometric functions, which makes the unit circle a powerful tool in solving problems involving angles.

Sine Function

The sine function is one of the primary trigonometric functions. It is based on the unit circle, where it corresponds to the y-coordinate of a point on the unit circle. For a given angle \(t\), \(\sin t\) is equal to the y-value of the corresponding point \(P(x, y)\) on the unit circle.
The sine function helps in capturing the vertical component of a circle's point and is useful in understanding periodic phenomena. In trigonometry, \(\sin t\) is widely used, especially in calculus, physics, and engineering, to describe wave-like processes. So, in the given problem by inserting the coordinates \( (\frac{3}{5}, \frac{4}{5}) \), we easily calculate \(\sin t = \frac{4}{5}\).

Cosine Function

The cosine function is closely related to the sine function and is another key concept in trigonometry. It represents the horizontal component of a point on the unit circle. Thus, for an angle \(t\), \(\cos t\) is equivalent to the x-coordinate at the point \(P(x, y)\).
Like the sine function, the cosine function is also periodic and crucial for waveform analyses. In our problem, with the point \((\frac{3}{5}, \frac{4}{5}) \) on the unit circle, the cosine value is easily identified: \(\cos t = \frac{3}{5}\). This function helps one to explore angular relationships and can also assist in proving trigonometric identities.

Tangent Function

The tangent function offers insight into the ratio of the sine and cosine functions. It can be seen as a slope of a line extending from the origin through a point on the unit circle. Mathematically, for an angle \(t\), \(\tan t\) is given by \( \frac{\sin t}{\cos t} \). This captures the idea of slope as it relates rise (sine) over run (cosine).
The tangent function is particularly useful when dealing with angles larger than \(90^\circ\) or when dealing with complex calculations involving angles. In the problem at hand, using \(\sin t = \frac{4}{5}\) and \(\cos t = \frac{3}{5}\), we calculate \(\tan t = \frac{4}{3}\). Understanding the tangent function aids in both theoretical and practical applications of trigonometry.