Problem 38 - 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 \(\tan t\). $$\left(\frac{24}{25},-\frac{7}{25}\right)$$

Step-by-Step Solution

Answer

\(\sin t = -\frac{7}{25}\), \(\cos t = \frac{24}{25}\), \(\tan t = -\frac{7}{24}\).

Step 1: Understand Terminal Point Coordinates

The terminal point is given as \( P\left( \frac{24}{25}, -\frac{7}{25} \right) \), where \( \frac{24}{25} \) represents \( \cos t \) and \( -\frac{7}{25} \) represents \( \sin t \).

Step 2: Identify \(\cos t\)

From the coordinates of the terminal point, we identify the \(x\)-coordinate as \( \frac{24}{25} \). Thus, \( \cos t = \frac{24}{25} \).

Step 3: Identify \(\sin t\)

Similarly, the \(y\)-coordinate of the terminal point is \( -\frac{7}{25} \). Thus, \( \sin t = -\frac{7}{25} \).

Step 4: Calculate \(\tan t\)

The tangent of \( t \) is the ratio of \( \sin t \) to \( \cos t \). So, calculate \( \tan t = \frac{\sin t}{\cos t} = \frac{-\frac{7}{25}}{\frac{24}{25}} = -\frac{7}{24} \).

Key Concepts

Understanding the Sine Function: sin t

Sine, represented as \( \sin t \), is one of the fundamental trigonometric ratios associated with a given angle \( t \). When dealing with a unit circle, the sine of an angle corresponds to the y-coordinate of its terminal point. This is because, within a unit circle, each point \((x, y)\) represents \( (\cos t, \sin t) \). The given terminal point is \( \left( \frac{24}{25}, -\frac{7}{25} \right) \), which tells us directly that \( \sin t = -\frac{7}{25} \).

• In any circle, the sine function helps us determine the vertical position of a point at a given angle.
• It is crucial for understanding wave patterns, oscillations, and many natural phenomena.
• Sine values are periodic, repeating every \( 2\pi \) radians or 360 degrees.

Being able to quickly identify \( \sin t \) from a terminal point helps when calculating other related trigonometric ratios.

Grasping the Cosine Function: cos t

The cosine function, noted as \( \cos t \), measures the horizontal position of the terminal point of an angle \( t \) on the unit circle. The cosine of an angle gives you the x-coordinate from a circle and is essential when examining distances and projections. In our example, the terminal point \( \left( \frac{24}{25}, -\frac{7}{25} \right) \) immediately reveals \( \cos t = \frac{24}{25} \).

• Cosine functions help determine spatial relationships like those in triangle measurements.
• Just like sine, cosine values repeat every \( 2\pi \) radians.
• Understanding \( \cos t \) is key when combining angles or finding sines of complementary angles.

Remember, \( \cos \) can illustrate how far horizontally a point stretches from the circle's center, which is vital across physics, engineering, and computer graphics.

Exploring the Tangent Function: tan t

Tangent, or \( \tan t \), is a bit different because it isn't tied directly to a circle's radius like sine or cosine. Instead, it\'s the ratio of sine to cosine, representing the slope of the line made by the angle \( t \). Simply put, \( \tan t = \frac{\sin t}{\cos t} \). Given our known values, we find \( \tan t = \frac{-\frac{7}{25}}{\frac{24}{25}} = -\frac{7}{24} \). This illustrates how steep or flat the angle \( t \) appears from the perspective of a unit circle.

• Tangent is crucial in calculus, especially concerning limits and derivatives.
• It finds use in real-world applications such as calculating inclines and heights.
• Being able to derive \( \tan t \) helps in resolving complex angle problems.

Understanding \( \tan t \) assists with interpreting changes in direction and predicting paths or trajectories.