Problem 74 - 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. $$P(-1,2)$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{2\sqrt{5}}{5}, \cos \theta = \frac{-\sqrt{5}}{5}, \tan \theta = -2, \csc \theta = \frac{\sqrt{5}}{2}, \sec \theta = -\sqrt{5}, \cot \theta = -\frac{1}{2} \).

Step 1: Identify the coordinates

The point \( P \) given is \( (-1, 2) \). This means \( x = -1 \) and \( y = 2 \).

Step 2: Calculate the radius (r)

The radius \( r \) is the distance from the origin \((0,0)\) to the point \((-1,2)\). Use the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

Step 3: Determine the sine function

The sine of \( \theta \) is defined as the ratio of the y-coordinate to the radius: \[ \sin \theta = \frac{y}{r} = \frac{2}{\sqrt{5}} \] To rationalize, multiply numerator and denominator by \( \sqrt{5} \): \[ \sin \theta = \frac{2\sqrt{5}}{5} \]

Step 4: Determine the cosine function

The cosine of \( \theta \) is defined as the ratio of the x-coordinate to the radius: \[ \cos \theta = \frac{x}{r} = \frac{-1}{\sqrt{5}} \] To rationalize, multiply numerator and denominator by \( \sqrt{5} \): \[ \cos \theta = \frac{-\sqrt{5}}{5} \]

Step 5: Determine the tangent function

The tangent of \( \theta \) is the ratio of the sine to the cosine, or the ratio of the y-coordinate to the x-coordinate: \[ \tan \theta = \frac{y}{x} = \frac{2}{-1} = -2 \]

Step 6: Determine the cosecant function

The cosecant is the reciprocal of the sine: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{5}}{2} \]

Step 7: Determine the secant function

The secant is the reciprocal of the cosine: \[ \sec \theta = \frac{1}{\cos \theta} = -\sqrt{5} \]

Step 8: Determine the cotangent function

The cotangent is the reciprocal of the tangent: \[ \cot \theta = \frac{1}{\tan \theta} = -\frac{1}{2} \]

Key Concepts

Understanding Sine Function in Trigonometry

Sine is one of the primary trigonometric functions and is often expressed as \( \sin \theta \). It represents the ratio of the opposite side to the hypotenuse in a right triangle. In the context of placing a point \( P(x, y) \) on the Cartesian plane, sine can be defined using the radius \( r \), which is the distance from the origin. The formula used is \( \sin \theta = \frac{y}{r} \). This ratio gives us an idea of how much vertical elevation is present with respect to the hypotenuse.

For example, in our problem, the point \( P(-1, 2) \) indicates \( x = -1 \) and \( y = 2 \). By calculating \( r \) using the formula \( \sqrt{x^2 + y^2} \), we find \( r = \sqrt{5} \). Therefore, \( \sin \theta = \frac{2}{\sqrt{5}} \) is the sine value, which when rationalized, becomes \( \frac{2\sqrt{5}}{5} \).

Understanding this concept helps us see how the vertical component of a point influences the sine value on a unit circle. This function is particularly useful in various fields like physics, engineering, and computer graphics, allowing us to model periodic phenomena.

Exploring the Cosine Function

The cosine function is another critical trigonometric function denoted as \( \cos \theta \). It models the ratio of the adjacent side over the hypotenuse in a right-angled triangle. In terms of coordinates on the Cartesian plane, it is calculated as the ratio of the x-coordinate to the radius \( r \). Thus, \( \cos \theta = \frac{x}{r} \).

For a point \( P(-1, 2) \), and radius \( r \) as \( \sqrt{5} \), the value becomes \( \cos \theta = \frac{-1}{\sqrt{5}} \). After rationalizing, we find \( \cos \theta = \frac{-\sqrt{5}}{5} \). This tells us about the horizontal displacement of the point in relation to the hypotenuse.

In practical applications, cosine is pivotal when calculating angles and modeling oscillations like waves and sound. Knowing the cosine value aids in numerous scenarios such as determining phase relationships in signals.

Grasping the Tangent Function

The tangent function is signified by \( \tan \theta \) and represents the ratio of the sine to cosine, or alternatively, the ratio of the opposite to the adjacent side of a triangle. Mathematically, \( \tan \theta = \frac{y}{x} \). It can be seen as the slope of the line connecting the origin to the point \( P(x, y) \).

In our specific example, at \( P(-1, 2) \), we find \( \tan \theta = \frac{2}{-1} = -2 \). This reflects the tilt or steepness of the line and provides us with important insights, especially in physics, where angles against a plane are crucial.

Tangent is widely used in calculus, geometry, and engineering. It can help solve problems involving slopes and inclines and is integral when calculating trajectories and paths. Mastery of tangent helps simplify complex angular relationships, making it a core concept in trigonometry.