Trigonometric Functions And Coordinates On Terminal Ray

When we talk about trigonometry, we often deal with angles and their relationships to the sides of triangles. However, trigonometry also extends beautifully into the coordinate plane, allowing us to describe points on a circle or a ray using trigonometric functions. Let's dive into how the sine, cosine, and tangent of an angle $\theta$ can help us pinpoint coordinates $(x, y)$ on the terminal ray of that angle. This is a fundamental concept that bridges geometry and algebra, offering a powerful way to visualize and calculate relationships in the plane.

Understanding the Terminal Ray and Trigonometric Functions

Imagine an angle $\theta$ in standard position within the Cartesian coordinate system. Its initial side lies along the positive x-axis, and its vertex is at the origin $(0,0)$. The terminal ray is the ray that results after rotating the initial side by the angle $\theta$. Now, consider any point $(x, y)$ that lies on this terminal ray, other than the origin itself. If we draw a line segment from the origin to this point $(x, y)$, we create a right triangle (or we can think of it in terms of the coordinates directly). The distance from the origin to the point $(x, y)$ is $r = \sqrt{x^2 + y^2}$. This distance $r$ is always positive. In this context, the trigonometric functions are defined as follows:

• Sine ($\sin \theta$): The ratio of the y-coordinate to the distance $r$. So, $\sin \theta = \frac{y}{r}$.
• Cosine ($\cos \theta$): The ratio of the x-coordinate to the distance $r$. So, $\cos \theta = \frac{x}{r}$.
• Tangent ($\tan \theta$): The ratio of the y-coordinate to the x-coordinate. So, $\tan \theta = \frac{y}{x}$ (provided $x \neq 0$).

These definitions are incredibly versatile because they work for any angle, not just those in the first quadrant or those that form acute angles within right triangles. They apply to angles that are positive, negative, or greater than 360 degrees, and they accurately describe the position of points on the terminal ray.

Applying the Definitions to a Specific Angle

Let's consider the specific information given: $\sin \theta = -\frac{77}{85}$, $\cos \theta = \frac{36}{85}$, and $\tan \theta = -\frac{77}{36}$. Our goal is to find the coordinates $(x, y)$ of a point on the terminal ray of angle $\theta$. We know from the definitions that $\cos \theta = \frac{x}{r}$ and $\sin \theta = \frac{y}{r}$.

From the given cosine value, we have $\frac{36}{85} = \frac{x}{r}$. This implies that $x$ and $r$ are in the ratio 36 to 85. Since $r$ represents a distance from the origin, it must be positive. Therefore, we can infer that $x = 36k$ and $r = 85k$ for some positive constant $k$. Similarly, from the sine value, we have $-\frac{77}{85} = \frac{y}{r}$. This suggests that $y$ and $r$ are in the ratio -77 to 85. Thus, $y = -77k$ and $r = 85k$ for the same positive constant $k$. Notice that both equations consistently give us $r=85k$.

We can also verify the tangent value using these relationships: $\tan \theta = \frac{y}{x} = \frac{-77k}{36k} = -\frac{77}{36}$, which matches the given information. This consistency confirms our approach.

Determining the Coordinates $(x, y)$

The key insight here is that the values of $\sin \theta$ and $\cos \theta$ directly relate the coordinates $(x, y)$ to the distance $r$. Specifically, we can rewrite the definitions as:

• $x = r \cos \theta$
• $y = r \sin \theta$

We are given $\cos \theta = \frac{36}{85}$ and $\sin \theta = -\frac{77}{85}$. If we choose a specific point on the terminal ray, we can simplify the problem. The simplest choice for $r$ that makes $x$ and $y$ integers (or at least simpler values) is to set $r$ equal to the common denominator of the sine and cosine values, provided that denominator is positive.

In our case, the common denominator is 85, which is positive. So, let's choose $r = 85$. This choice corresponds to setting our scaling factor $k=1$ from the previous step.

Using this value of $r$, we can find $x$ and $y$:

$x = r \cos \theta = 85 \times \frac{36}{85} = 36$

$y = r \sin \theta = 85 \times -\frac{77}{85} = -77$

Therefore, the coordinates of a point $(x, y)$ on the terminal ray of angle $\theta$ are $(36, -77)$. It's important to note that this is just one such point. Any point $(x, y)$ on the terminal ray will be of the form $(36k, -77k)$ for any positive real number $k$. However, when asked for the coordinates, it's conventional to provide the simplest representation where $r$ is the positive denominator or the least common multiple of denominators if they differ, leading to integer coordinates if possible.

Quadrant Analysis and Consistency Check

Let's perform a quick check to ensure our coordinates make sense. We are given $\cos \theta = \frac{36}{85}$ (positive) and $\sin \theta = -\frac{77}{85}$ (negative). In the Cartesian plane:

• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y < 0

Since $x$ is positive and $y$ is negative, the terminal ray of angle $\theta$ must lie in Quadrant IV. Our calculated point $(36, -77)$ has a positive x-coordinate and a negative y-coordinate, which is consistent with Quadrant IV. This provides strong validation for our derived coordinates.

Furthermore, the tangent function, $\tan \theta = \frac{y}{x}$, is negative in both Quadrant II and Quadrant IV. Our calculation $\tan \theta = -\frac{77}{36}$ is indeed negative. When combined with the signs of sine and cosine, the quadrant is uniquely determined. Cosine positive and sine negative definitively places the angle in Quadrant IV.

The Significance of $r$

The value of $r = \sqrt{x^2 + y^2}$ is crucial. In our case, with the point $(36, -77)$, the distance $r$ is:

$r = \sqrt{36^2 + (-77)^2}$ $r = \sqrt{1296 + 5929}$ $r = \sqrt{7225}$

To find the square root of 7225, we can recognize that numbers ending in 25 often have square roots ending in 5. Let's try numbers like 85:

$85 \times 85 = (80+5)(80+5) = 6400 + 400 + 400 + 25 = 7225$.

So, $r = 85$. This confirms that our chosen point $(36, -77)$ results in a distance $r=85$ from the origin, which perfectly matches the denominators in the given sine and cosine values. This is why choosing $r$ to be the common positive denominator is often the most straightforward path to finding a specific point's coordinates.

Conclusion: Connecting Trigonometry and Coordinates

In essence, the trigonometric functions $\sin \theta$ and $\cos \theta$ provide the ratios that define the position of any point on the terminal ray relative to its distance from the origin. By understanding that $x = r \cos \theta$ and $y = r \sin \theta$, we can determine coordinates for any point on that ray. The specific values $\sin \theta = -\frac{77}{85}$ and $\cos \theta = \frac{36}{85}$ directly inform us that for every 85 units of distance from the origin along the terminal ray, the x-coordinate increases by 36 units and the y-coordinate decreases by 77 units. By selecting the simplest positive distance, $r=85$, we readily find the coordinates $(36, -77)$ for a point on the terminal ray of angle $\theta$. This powerful relationship is a cornerstone of trigonometry's application in analytical geometry.

For further exploration into the unit circle and trigonometric functions, you can visit Khan Academy's mathematics section or Brilliant.org's mathematics courses.