Problem 72 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The six trigonometric functions are: \( \sin \theta = -\frac{15}{17}, \cos \theta = -\frac{8}{17}, \tan \theta = \frac{15}{8}, \csc \theta = -\frac{17}{15}, \sec \theta = -\frac{17}{8}, \cot \theta = \frac{8}{15} \).

Step 1: Find the Radius (r)

To find the six trigonometric functions, first compute the radius (r), which acts like the hypotenuse of the right triangle formed by the x and y coordinates. Use the formula: \[ r = \sqrt{x^2 + y^2} \]Here, \(x = -8\) and \(y = -15\). Calculate:\[ r = \sqrt{(-8)^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \]

Step 2: Calculate Sine (sin θ)

Sine is defined as the ratio of the opposite side to the hypotenuse in a right triangle. For point \(P(-8, -15)\), the opposite side is the y-coordinate:\[ \sin \theta = \frac{y}{r} = \frac{-15}{17} \]

Step 3: Calculate Cosine (cos θ)

Cosine is the ratio of the adjacent side (x-coordinate) to the hypotenuse:\[ \cos \theta = \frac{x}{r} = \frac{-8}{17} \]

Step 4: Calculate Tangent (tan θ)

Tangent is the ratio of the opposite side to the adjacent side:\[ \tan \theta = \frac{y}{x} = \frac{-15}{-8} = \frac{15}{8} \]

Step 5: Calculate Cosecant (csc θ)

Cosecant is the reciprocal of sine:\[ \csc \theta = \frac{r}{y} = \frac{17}{-15} = -\frac{17}{15} \]

Step 6: Calculate Secant (sec θ)

Secant is the reciprocal of cosine:\[ \sec \theta = \frac{r}{x} = \frac{17}{-8} = -\frac{17}{8} \]

Step 7: Calculate Cotangent (cot θ)

Cotangent is the reciprocal of tangent:\[ \cot \theta = \frac{x}{y} = \frac{-8}{-15} = \frac{8}{15} \]

Key Concepts

Sine

The sine function, abbreviated as \( \sin \theta \), is one of the fundamental trigonometric functions. It represents the ratio of the length of the side opposite to the angle \( \theta \) to the length of the hypotenuse in a right-angled triangle.
In our specific exercise with point \( P(-8, -15) \) on the Cartesian plane, the y-coordinate \( -15 \) serves as the opposite side, and the radius \( r = 17 \) we calculated earlier serves as the hypotenuse.
The formula becomes:

• \( \sin \theta = \frac{-15}{17} \)

This ratio tells us how tall the triangle is in comparison to its diagonal in the context of our problem.
This concept is essential for analyzing periodic phenomena, like sound waves and light signals, through their sinusoidal representation.

Cosine

Cosine, written as \( \cos \theta \), is another basic trigonometric function describing ratios in right triangles. Specifically, it relates the adjacent side to the hypotenuse.
For the point \( P(-8, -15) \) on the terminal side of the angle, the adjacent side is given by the x-coordinate \( -8 \).
This leads to the expression:

• \( \cos \theta = \frac{-8}{17} \)

By analyzing this ratio, we gain insight into how the triangle's base compares to its hypotenuse. Cosine values help determine projections and are vital in calculating components of vectors in physics and engineering.
Exploring cosine allows us to solve many real-life problems involving distances and angles.

Tangent

Tangent, signified as \( \tan \theta \), is defined as the ratio of the opposite side to the adjacent side within a right triangle. In our context, it expresses the steepness of the slope formed by the line through \( P(-8, -15) \).
With our coordinates, the tangent is formulated as:

• \( \tan \theta = \frac{-15}{-8} = \frac{15}{8} \)

This simplification arises as both x and y coordinate are negative, making their ratio positive. Understanding the tangent function is crucial for describing slopes in geometry and calculating inclines in real-world applications, like road design and architecture.
Tangible applications of tangent include determining elevation changes over a horizontal distance or predicting stone trajectories in projectile motion.

Cosecant

Cosecant, abbreviated \( \csc \theta \), is the reciprocal of sine. This function comes in handy when dealing with problems where reciprocal values are more convenient or necessary.
For point \( P(-8, -15) \), the cosecant evaluates to:

• \( \csc \theta = \frac{17}{-15} = -\frac{17}{15} \)

The emergence of negative values directly influences the direction of the trigonometric angle and provides context to the respective quadrant of the angle.
Observing the cosecant function helps in solving equations that involve large ratios or inverses, proving valuable in fields like electrical engineering and physics.

Secant

Secant, or \( \sec \theta \), is the reciprocal to cosine. This function exhibits how a base length of a triangle extends concerning the whole triangle's hypotenuse.
Derived from point \( P(-8, -15) \), secant computes as:

• \( \sec \theta = \frac{17}{-8} = -\frac{17}{8} \)

With this negative ratio, secant tells us about the directions in which the sides extend, particularly essential in understanding phenomena with opposite-facing components.
Secant is particularly useful in solving problems involving light refraction and lens making, where understanding inversions between lengths provides solutions.

Cotangent

Cotangent, denoted as \( \cot \theta \), represents the reciprocal of the tangent function. In simpler terms, it reverses the steepness analysis provided by tangent.
Applied to our existing point \( P(-8, -15) \), the cotangent derives as:

• \( \cot \theta = \frac{-8}{-15} = \frac{8}{15} \)

Understanding cotangent assists in grasping how steep a slope is when approached from a complementary angle.
Cotangent values are indispensable for calculations involving relationships such as angles between clock hands and heights of distant objects via indirect measurements. It shows the flipping view of an angle's slope, offering distinct insights in calculus and advancing geometric proofs.