Problem 72 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and \(P\) is on the terminal side. $$P(-8,-15)$$

Step-by-Step Solution

Answer

\(\sin(\theta) = -\frac{15}{17}, \cos(\theta) = -\frac{8}{17}, \tan(\theta) = \frac{15}{8}, \csc(\theta) = -\frac{17}{15}, \sec(\theta) = -\frac{17}{8}, \cot(\theta) = \frac{8}{15}\."

Step 1: Understand the Problem

We are given a point \(P(-8, -15)\) on the terminal side of angle \(\theta\) in standard position. We need to find the exact values of the six trigonometric functions for \(\theta\).

Step 2: Find the Distance (r)

In trigonometry, the distance \(r\) from the origin to the point \(P(x, y)\) is important. It can be calculated using the distance formula.\[ r = \sqrt{x^2 + y^2} \]Substituting the values \(x = -8\) and \(y = -15\), we find: \[ r = \sqrt{(-8)^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \].

Step 3: Calculate Sine and Cosecant

The sine of \(\theta\) is determined by the ratio \( \frac{y}{r} \). Thus:\[ \sin(\theta) = \frac{-15}{17}\]Cosecant is the reciprocal of sine:\[\csc(\theta) = \frac{17}{-15}\].

Step 4: Calculate Cosine and Secant

The cosine of \(\theta\) is determined by the ratio \( \frac{x}{r} \). Thus:\[\cos(\theta) = \frac{-8}{17}\]Secant is the reciprocal of cosine:\[\sec(\theta) = \frac{17}{-8}\].

Step 5: Calculate Tangent and Cotangent

The tangent of \(\theta\) is determined by the ratio \( \frac{y}{x} \). Thus:\[\tan(\theta) = \frac{-15}{-8} = \frac{15}{8}\]Cotangent is the reciprocal of tangent:\[\cot(\theta) = \frac{8}{15}\].

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving angles and distances in triangles. The primary focus is on the six fundamental trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These functions describe ratios of a right triangle's sides relative to its angles.
\(\sin(\theta)\), \(\cos(\theta)\), and \(\tan(\theta)\) are the basic trigonometric functions, referring to ratios of side lengths in a right triangle:

• Sine (sin): Opposite side over hypotenuse, \(\sin(\theta) = \frac{y}{r}\).
• Cosine (cos): Adjacent side over hypotenuse, \(\cos(\theta) = \frac{x}{r}\).
• Tangent (tan): Opposite side over adjacent side, \(\tan(\theta) = \frac{y}{x}\).

Trigonometry allows you to explore patterns and solve real-life problems in fields like physics and engineering. Understanding these foundational concepts is crucial for navigating more complex applications later.

Standard Position

An angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies on the positive x-axis. This means it starts at 0 degrees or radians and measures counterclockwise.
Being in standard position helps define the terminal side — where the angle ends and thus relates to a specific coordinate point like \(P(-8, -15)\) in this exercise. This position assists in calculating trigonometric functions readily based on the coordinate and the radial distance from the origin.

• The x-coordinate represents horizontal movement.
• The y-coordinate signifies vertical movement.

The location of the point identifies the angle's quadrant, which influences the signs of trigonometric functions. For example, a terminal point in the third quadrant (like \(-8, -15\)) implies both sine and cosine are negative.

Reciprocal Functions

Reciprocal functions are derived from the basic trigonometric functions and represent inverse relationships. Understanding these reciprocals is essential to fully grasp the breadth of trigonometric applications. They are often used in situations where direct measurement is not possible, and they provide important computational functions:

• Cosecant (csc): The reciprocal of sine, \(\csc(\theta) = \frac{1}{\sin(\theta)}\).
• Secant (sec): The reciprocal of cosine, \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
• Cotangent (cot): The reciprocal of tangent, \(\cot(\theta) = \frac{1}{\tan(\theta)}\).

These functions play a crucial role when angles are derived from arc lengths on the unit circle, associating every angle with a unique point. Additionally, reciprocal functions ensure the symmetry and periodic nature of trigonometric graphs can be described in a reliable and repeatable manner.