Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-10,\) the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c.tan \(\theta\) d. \(\sec \theta\) e. \(\csc \theta\) f. \(\cot \theta\) $$ P\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) $$

Step-by-Step Solution

Answer

a. \( \sin \theta = \frac{\sqrt{3}}{2} \); b. \( \cos \theta = -\frac{1}{2} \); c. \( \tan \theta = -\sqrt{3} \); d. \( \sec \theta = -2 \); e. \( \csc \theta = \frac{2\sqrt{3}}{3} \); f. \( \cot \theta = -\frac{\sqrt{3}}{3} \).

Step 1: Identify the Coordinates and Determine the Quadrant

Given that the terminal side of \( \angle ROP \) intersects the unit circle at \( P \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \), we determine which quadrant this point is located in. Here, the x-coordinate is negative and the y-coordinate is positive; therefore, the point is in the second quadrant.

Step 2: Determine \( \sin \theta \) from Point P

Since point \( P \) lies on the unit circle, \( \sin \theta \) is equal to the y-coordinate of the point. Therefore, \( \sin \theta = \frac{\sqrt{3}}{2} \).

Step 3: Determine \( \cos \theta \) from Point P

Similarly, \( \cos \theta \) is equal to the x-coordinate of point \( P \). Hence, \( \cos \theta = -\frac{1}{2} \).

Step 4: Calculate \( \tan \theta \)

The tangent of an angle is the ratio of the sine to the cosine for that angle. Thus \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \).

Step 5: Calculate \( \sec \theta \)

The secant of an angle is the reciprocal of the cosine. Therefore, \( \sec \theta = \frac{1}{\cos \theta} = -2 \).

Step 6: Calculate \( \csc \theta \)

The cosecant of an angle is the reciprocal of the sine. Therefore, \( \csc \theta = \frac{1}{\sin \theta} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \) after rationalizing the denominator.

Step 7: Calculate \( \cot \theta \)

The cotangent of an angle is the reciprocal of the tangent. Thus, \( \cot \theta = \frac{1}{\tan \theta} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Key Concepts

Unit Circle

The unit circle is a critical concept in trigonometry. It represents a circle with a radius of exactly one unit. This circle is centered at the origin of a coordinate plane, making it a perfect illustration for understanding trigonometric functions. Every point on the unit circle has coordinates that are directly related to the sine and cosine of the angles they represent.

Using coordinates to represent points on the unit circle simplifies calculations since if any point \(P(x, y)\), lies on the unit circle, \(x^2 + y^2 = 1\). This relationship helps confirm that the radius is always 1.

Additionally, the angle at which a line from the origin intersects a point on the unit circle can help determine which quadrant the point is located in. This information is vital when calculating different trigonometric ratios.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that derive from the unit circle. By positioning any angle \(\theta\) in the unit circle, the sine of \(\theta\) will equate to the y-coordinate of the point where the terminal side of the angle intersects the circle. Conversely, the cosine of that angle corresponds to the x-coordinate.

This relationship simplifies the determination of these functions without the need for additional computations. When given specific coordinates from the unit circle, in our example \(P\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), we immediately identify that \ \sin \theta = \frac{\sqrt{3}}{2}\ \ and \ \cos \theta = -\frac{1}{2}\.

This provides a direct way to extract sine and cosine values, offering a simplified understanding and quicker calculations for these trigonometric functions.

Tangent and Cotangent

Tangent and cotangent are two more trigonometric functions that express relationships between sine and cosine. The tangent of an angle \(\theta\) on the unit circle is found by dividing the sine of the angle by the cosine. It represents the slope of the line generated by the angle.

In our specific example: \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \).

Cotangent, however, is the reciprocal of tangent. It is a useful function when considering angles that might cause a tangent function to be undefined. Switching roles, it is calculated as:

• \( \cot \theta = \frac{1}{\tan \theta} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \)

By rationalizing cotangent, we avoid irrational denominators, a standard mathematical practice.

Secant and Cosecant

Secant and cosecant are reciprocals of the cosine and sine functions, respectively. They extend our capacity to relate angles to their trigonometric attributes beyond the typical sine and cosine.

For an angle \(\theta\) where you already know the cosine, calculating the secant involves taking the reciprocal:

• \( \sec \theta = \frac{1}{\cos \theta} = -2 \)

Similarly, the cosecant is derived from the sine:

• \( \csc \theta = \frac{1}{\sin \theta} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \)

Rationalizing can make these expressions cleaner, an important step if further calculations are needed.

These alternate forms help us better understand scenarios where sine or cosine are zero, leading to undefined results and emphasizing the importance of these alternatives in trigonometry.