Problem 8 - Algebra 2

Question

The given point \(P\) is located on the unit circle. Find \(\sin \theta\) and \(\cos \theta\) $$ P\left(-\frac{12}{13},-\frac{5}{13}\right) $$

Step-by-Step Solution

Answer

\(\sin \theta = -\frac{5}{13}\), \(\cos \theta = -\frac{12}{13}\).

Step 1: Understanding the Unit Circle

On the unit circle, a point \((x, y)\) represents \((\cos \theta, \sin \theta)\). The point given is \(P\left(-\frac{12}{13}, -\frac{5}{13}\right)\). Here, \(x = -\frac{12}{13}\) and \(y = -\frac{5}{13}\).

Step 2: Identifying \(\cos \theta\)

Since the \(x\)-coordinate of the point \(P\) on the unit circle is equal to \(\cos \theta\), we have \(\cos \theta = -\frac{12}{13}\).

Step 3: Identifying \(\sin \theta\)

Since the \(y\)-coordinate of the point \(P\) on the unit circle is equal to \(\sin \theta\), we have \(\sin \theta = -\frac{5}{13}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, especially when dealing with angles and distances on the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle helps in understanding how the sine and cosine functions work.The main trigonometric functions include:

• Sine (\(\sin \theta\)) - represents the vertical coordinate (y-value) on the unit circle.
• Cosine (\(\cos \theta\)) - represents the horizontal coordinate (x-value) on the unit circle.
• Tangent (\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)) - a ratio of sine and cosine.

On the unit circle, any coordinate point \( (x, y)\) corresponds to \((\cos \theta, \sin \theta)\). This means that for any angle \(\theta\), you can find its sine and cosine directly by looking at the point's coordinates on the unit circle.Understanding this relationship is key to mastering trigonometry and solving various mathematical problems that involve angles and distances.

Cosine

Cosine is one of the fundamental trigonometric functions. It is associated with the x-coordinate of a point on the unit circle. For example, if you have a point \((x, y)\) on the unit circle, the x-coordinate \(x\) is the value of \(\cos \theta\) for that point.

• In the unit circle, as \(\theta\) varies, \(\cos \theta\) takes values between -1 and 1.
• \(\cos \theta\) starts at 1 when \(\theta = 0\), representing the point (1, 0).
• It decreases to 0 when \(\theta = \frac{\pi}{2}\) (90 degrees), where the point is (0, 1).
• \(\cos \theta\) goes to -1 at \(\theta = \pi\) (180 degrees), corresponding to the point (-1, 0).
• Finally, it returns to 1 at \(\theta = 2\tau\) (360 degrees).

For the specific exercise point \(-\frac{12}{13}\,\), it tells us that the x-coordinate, or cosine value, at this angle on the unit circle is less than zero, indicating an angle in the second quadrant where cosine is typically negative.

Sine

The sine function is another core trigonometric function, representing the y-coordinate of a point on the unit circle. In any point \((x, y)\)on the unit circle, the y-coordinate (y) is the value of \(\sin \theta\) for that particular angle.

• Just like \(\cos \theta\), \(\sin \theta\) values range from -1 to 1.
• When \(\theta = 0\), sine starts at 0, corresponding to the point (1, 0).
• It reaches 1 at \(\theta = \frac{\pi}{2}\) (90 degrees), which is the topmost point (0, 1).
• As \(\theta\)increases to \(\pi\) (180 degrees), \(\sin \theta\) decreases back to 0 at (-1, 0).
• It then goes down to -1 at \(\theta = \frac{3\pi}{2}\) (270 degrees), and returns to 0 at \(\theta = 2\tau\) (360 degrees), completing a full cycle.

For the point in our exercise, \(-\frac{5}{13}\,\) represents a negative sine value, showing that the point lies below the horizontal axis. This corresponds to an angle in the fourth quadrant, where sine values are negative.