Problem 36 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{24}{25},-\frac{7}{25}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify coordinates

The terminal point given is \( P \left( \frac{24}{25}, -\frac{7}{25} \right) \). From this, \( x = \frac{24}{25} \) and \( y = -\frac{7}{25} \).

Step 2: Use the unit circle identity

On the unit circle, the relationship is \( x = \cos t \) and \( y = \sin t \). Thus, \( \cos t = \frac{24}{25} \) and \( \sin t = -\frac{7}{25} \).

Step 3: Calculate tangent

The tangent of an angle \( t \) is given by \( \tan t = \frac{\sin t}{\cos t} \). Substitute the values: \( \tan t = \frac{-\frac{7}{25}}{\frac{24}{25}} = -\frac{7}{24} \).

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry. It allows us to understand the relationship between angles and the coordinates of points on a circle. A unit circle is a circle with a radius of one, centered at the origin of a coordinate plane.

• The circumference of the unit circle is learning ground for understanding sine, cosine, and tangent functions.
• The angle in the unit circle is measured in radians, where one full rotation around the circle equals to \(2\pi\) radians.
• Coordinates on the unit circle correspond to the cosine and sine of the angle \(t\).

The standard equation for the unit circle is \(x^2 + y^2 = 1\). Any point \((x, y)\) on the unit circle satisfies this equation. This relationship helps establish that cosine and sine values can never be more significant than one. Using these coordinates, we can easily find the sine, cosine, and tangent for any angle as it correlates to the points on this circle.

Sine and Cosine

Sine and cosine functions are fundamental trigonometric functions describing a right triangle's angles and side ratios. They are defined as the coordinates \((x, y)\) of a point on the unit circle for a given angle.- **Cosine** represents the horizontal position (the \(x\)-coordinate) of a point on the unit circle.- **Sine** represents the vertical position (the \(y\)-coordinate) of the same point.

Understanding Cosine

For the angle \(t\), the cosine is given by the circumference position on the x-axis, \(\cos t = x\). In the exercise, the cosine valued at \(t\) was \(\frac{24}{25}\), indicating a near-complete rotation since the maximum value cosine can be is 1.0.

Understanding Sine

Similarly, sine evaluates how far above or below the x-axis the point lies. The sine for this angle \(t\) is \(\sin t = y = -\frac{7}{25}\), showing the downward direction since it's negative, hinting the point is below the x-axis.These trigonometric identities are crucial as they form the foundation for understanding how angles translate into coordinates on a circle.

Tangent

The tangent function offers a different perspective compared to sine and cosine, focusing on the ratio between them. It can be thought of as describing "how steep" a line is that connects the origin to a point on the unit circle.

• Tangent is defined as \(\tan t = \frac{\sin t}{\cos t}\).
• This ratio expresses the slope of the line formed to the point \((x,y)\).

From the exercise, - We calculated \(\tan t = \frac{-\frac{7}{25}}{\frac{24}{25}}\), which simplifies to \(-\frac{7}{24}\).This implies that the line is leaning downwards due to the negative value, matching its position on the Cartesian plane. Tangent functions help in understanding angles as intersections of straight lines, representing a significant bridge between linear equations and circular trigonometric functions.