Problem 9 - 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. \(\cot \theta\) $$ \left(-\frac{\sqrt{7}}{5}, \frac{3 \sqrt{2}}{5}\right) $$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{3 \sqrt{2}}{5} \), \( \cos \theta = -\frac{\sqrt{7}}{5} \), \( \tan \theta = -\frac{3 \sqrt{14}}{7} \), \( \sec \theta = -\frac{5 \sqrt{7}}{7} \), \( \csc \theta = \frac{5 \sqrt{2}}{6} \), \( \cot \theta = -\frac{\sqrt{14}}{6} \).

Step 1: Identify coordinates

Identify the coordinates given where the terminal side intersects the unit circle: \( P = \left( -\frac{\sqrt{7}}{5}, \frac{3 \sqrt{2}}{5} \right) \). These coordinates directly give the values for \( \cos \theta \) and \( \sin \theta \).

Step 2: Determine \(\sin \theta\)

The y-coordinate of intersection on the unit circle gives the value of \( \sin \theta \). So, \( \sin \theta = \frac{3 \sqrt{2}}{5} \).

Step 3: Determine \(\cos \theta\)

The x-coordinate of intersection on the unit circle gives the value of \( \cos \theta \). Thus, \( \cos \theta = -\frac{\sqrt{7}}{5} \).

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

\( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3 \sqrt{2}}{5}}{-\frac{\sqrt{7}}{5}} = -\frac{3 \sqrt{2}}{\sqrt{7}} = -\frac{3 \sqrt{14}}{7} \) after rationalizing the denominator.

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

\( \sec \theta \) is the reciprocal of \( \cos \theta \), so \( \sec \theta = -\frac{5}{\sqrt{7}} = -\frac{5 \sqrt{7}}{7} \) after rationalizing the denominator.

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

\( \csc \theta \) is the reciprocal of \( \sin \theta \), so \( \csc \theta = \frac{5}{3 \sqrt{2}} = \frac{5 \sqrt{2}}{6} \) after rationalizing the denominator.

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

\( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{\sqrt{7}}{5}}{\frac{3 \sqrt{2}}{5}} = -\frac{\sqrt{7}}{3 \sqrt{2}} = -\frac{\sqrt{14}}{6} \) after rationalizing the denominator.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the relationships between angles and sides of right triangles.
They are also critical when working with the unit circle, where each angle corresponds to specific values of sine, cosine, and other functions.

• Sine (\( ext{sin} \theta\)): This function measures the vertical component of a point on the unit circle.
• Cosine (\( ext{cos} \theta\)): Indicates the horizontal component.
• Tangent (\( ext{tan} \theta\)): Represents the ratio of sine to cosine, \( an \theta = \frac{\sin \theta}{\cos \theta}\).

These functions allow us to model periodic phenomena and solve various practical problems in physics and engineering.

Sine and Cosine

Sine and cosine are perhaps the most well-known trigonometric functions. They offer insights into the angles in a periodic context.
They are directly related to the coordinates of a point on the unit circle. To find these values, simply identify the coordinates of the intersection point for an angle \(\theta\).

• The y-coordinate of the point gives you \( ext{sin} \theta\).
• The x-coordinate provides \( ext{cos} \theta\).

For example, given \( ext{P} = \left(-\frac{\sqrt{7}}{5}, \frac{3 \sqrt{2}}{5}\right)\),the readings are \(\text{cos} \theta = -\frac{\sqrt{7}}{5}\) and \(\text{sin} \theta = \frac{3 \sqrt{2}}{5}\).
This approach is essential for comprehending angles measured from the positive x-axis.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions complete the scene for unit circle problems.
They include secant, cosecant, and cotangent, which are the reciprocals of cosine, sine, and tangent, respectively.

• Secant (\( ext{sec} \theta\)): Defined as \(\text{sec} \theta = \frac{1}{\cos \theta}\)
• Cosecant (\( ext{csc} \theta\)): Defined as \(\text{csc} \theta = \frac{1}{\sin \theta}\)
• Cotangent (\( ext{cot} \theta\)): Defined as \(\text{cot} \theta = \frac{1}{\tan \theta}\)

These definitions are derived from their respective base functions. They are particularly useful for computations where reciprocal values are easier to work with or offer additional insight about angle properties on the unit circle.

Angle in Standard Position

An angle in standard position has its vertex at the origin of a coordinate system. The initial side of the angle lies along the positive x-axis.

The terminal side is where the angle stops rotating counter-clockwise.
For the unit circle, every angle can be represented by a terminal point, where its coordinates relate directly to specific values of trigonometric functions.

• The angle's measure is positive if rotated counter-clockwise, and negative if it is clockwise.
• This representation helps in straightforwardly evaluating trigonometric values for angles based on the x and y coordinates of the terminal point.

Understanding this concept of angles helps visualize and solve diverse problems in trigonometry, aiding the connection between geometric and algebraic representations.