Problem 18 - Algebra 2

Question

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((0,-6)\)

Step-by-Step Solution

Answer

\(\sin(\theta) = -1\), \(\cos(\theta) = 0\), \(\tan(\theta) \) is undefined; \(\csc(\theta) = -1\), \(\sec(\theta) \) is undefined, \(\cot(\theta) \) is undefined.

Step 1: Understanding the Position of Point and Angle

The terminal side of the angle \( \theta \) passes through the point \((0, -6)\). This means that \( \theta \) is on the negative y-axis, pointing downward.

Step 2: Determine the Basic Trigonometric Ratios

The point \((0, -6)\) represents a vertical line where the x-coordinate is 0, so \( r = \sqrt{0^2 + (-6)^2} = \sqrt{36} = 6 \). Thus, the hypotenuse \( r \) is 6.

Step 3: Calculate Sine

Sine of an angle is defined as \( \sin(\theta) = \frac{y}{r} \). Here, \( y = -6 \) and \( r = 6 \), so \( \sin(\theta) = \frac{-6}{6} = -1 \).

Step 4: Calculate Cosine

Cosine of an angle is defined as \( \cos(\theta) = \frac{x}{r} \). Here, \( x = 0 \) and \( r = 6 \), so \( \cos(\theta) = \frac{0}{6} = 0 \).

Step 5: Calculate Tangent

Tangent is defined as \( \tan(\theta) = \frac{y}{x} \). Here, \( y = -6 \) and \( x = 0 \). Since \( x = 0 \), tangent is undefined (division by zero).

Step 6: Calculate Cosecant

Cosecant is the reciprocal of sine, \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Since \( \sin(\theta) = -1 \), \( \csc(\theta) = -1 \).

Step 7: Calculate Secant

Secant is the reciprocal of cosine, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Since \( \cos(\theta) = 0 \), secant is undefined because you cannot divide by zero.

Step 8: Calculate Cotangent

Cotangent is the reciprocal of tangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Since tangent is undefined, similarly cotangent is also undefined.

Key Concepts

Understanding Sine

Sine is one of the fundamental trigonometric functions used to relate angles and sides in right triangles. It specifically measures the ratio of the length of the opposite side to the hypotenuse in a right triangle. The formula is given by:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the context of the exercise and the given point \((0, -6)\), the sine of \(\theta\) is calculated as \( \sin(\theta) = \frac{-6}{6} = -1 \).
This tells us that for an angle \(\theta\) whose terminal side points directly down along the negative y-axis, the sine value represents a full negative unit.
Sine is a crucial identifier in trigonometry because it defines the basic height of the triangle formed with respect to the y-axis direction. Understanding sine as a function of an angle helps uncover more intrinsic properties of circles and oscillatory motions.

Understanding Cosine

Cosine complements sine as another key trigonometric function. It is all about the relationship between the adjacent side and the hypotenuse in a right triangle. The cosine of angle \(\theta\) can be found using:

• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

For our example with point \((0, -6)\), the cosine is calculated as \( \cos(\theta) = \frac{0}{6} = 0 \).
The result \(0\) highlights how the horizontal component (or x-axis movement) is nonexistent for the given angle. As such, cosine reflects the zero level of horizontal projection when an angle points straight down. This concept is integral in solving various problems about range, projections, and even electricity waves.

Understanding Tangent

Tangent represents yet another trigonometric function, defined as the ratio of sine to cosine, or mathematically as the opposite over the adjacent. The expression for tangent is:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In simpler terms, it reveals how much vertical change occurs per unit of horizontal change.
In this particular problem, since the x-coordinate \(x = 0\), the tangent becomes undefined due to division by zero. This teaches us a pivotal lesson: when there's no horizontal direction like at points along the y-axis, tangent cannot be determined.
Students must understand that tangent provides insights into slopes, inclines, and changes in various contexts like engineering and physics fields.

Understanding Reciprocal Trigonometric Functions

The reciprocals of sine, cosine, and tangent lead to three additional trigonometric functions: cosecant, secant, and cotangent.

• Cosecant is the reciprocal of sine, represented as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). From the exercise, given \( \sin(\theta) = -1 \), the cosecant becomes \( \csc(\theta) = -1 \).
• Secant is the reciprocal of cosine, expressed as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). However, since \( \cos(\theta) = 0 \), the secant is undefined as you cannot divide by zero.
• Cotangent is the reciprocal of tangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \). As tangent is undefined due to division by zero, so is cotangent.

Reciprocal functions broaden the scope of problems we can address in trigonometry, particularly when describing waves, voltages, and periodic phenomena. Their values can also signify asymptotic behaviors, aiding in mastering calculus and other advanced mathematics.