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

Step-by-Step Solution

Answer

Sine: \( \frac{2\sqrt{5}}{5} \), Cosine: \( \frac{-\sqrt{5}}{5} \), Tangent: \(-2\), Cotangent: \(-\frac{1}{2}\), Secant: \(-\sqrt{5}\), Cosecant: \(\sqrt{5}\).

Step 1: Identify the Given Point

We are given the coordinates of point \( P(-1, 2) \). This means that \( x = -1 \) and \( y = 2 \) in the coordinate plane.

Step 2: Calculate the Distance from Origin - r

To find the value of \( r \), which is the distance from the origin to the point \( P \), we use the distance formula: \[ r = \sqrt{x^2 + y^2} \]Substituting the given values:\[ r = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

Step 3: Calculate Sine and Cosine

The sine of \( \theta \) is the ratio of \( y \) to \( r \):\[ \sin(\theta) = \frac{y}{r} = \frac{2}{\sqrt{5}} \]The cosine of \( \theta \) is the ratio of \( x \) to \( r \):\[ \cos(\theta) = \frac{x}{r} = \frac{-1}{\sqrt{5}} \]

Step 4: Calculate Tangent and Cotangent

The tangent of \( \theta \) is the ratio of \( y \) to \( x \):\[ \tan(\theta) = \frac{y}{x} = \frac{2}{-1} = -2 \]The cotangent is the reciprocal of tangent:\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{-1}{2} \]

Step 5: Calculate Secant and Cosecant

The secant of \( \theta \) is the reciprocal of cosine:\[ \sec(\theta) = \frac{1}{\cos(\theta)} = -\sqrt{5} \]The cosecant is the reciprocal of sine:\[ \csc(\theta) = \frac{1}{\sin(\theta)} = \sqrt{5} \]

Step 6: Simplify Trigonometric Functions

Rationalize the denominators where necessary. For the sine and cosine functions, we rationalize:\[ \sin(\theta) = \frac{2\sqrt{5}}{5} \]\[ \cos(\theta) = \frac{-\sqrt{5}}{5} \]

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry is a foundational concept in mathematics, allowing us to explore relationships between angles and side lengths in triangles. A right triangle has one 90-degree angle, and the sides are labeled relative to another angle, namely the opposite, adjacent, and hypotenuse. These sides are used to define six trigonometric functions:

• Sine (\( \sin \theta \) ) is the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \) ) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \) ) is the ratio of the opposite side to the adjacent side.
• Cosecant (\( \csc \theta \) ) is the reciprocal of sine.
• Secant (\( \sec \theta \) ) is the reciprocal of cosine.
• Cotangent (\( \cot \theta \) ) is the reciprocal of tangent.

In right triangle trigonometry, these ratios remain constant for a given angle, allowing us to solve for unknown side lengths when at least one side length and an angle are known. For example, if you know the length of the hypotenuse and the angle, you can find the opposite side using sine: opposite = hypotenuse × sine. Ensuring you understand these basis trigonometric relationships is crucial for moving forward in geometry and calculus.

Ratios and Proportions

Ratios and proportions are essential mathematical concepts that describe how quantities relate to each other. A ratio is a comparative value between two or more quantities, indicating how many times one value contains or is contained within the other. For instance, in trigonometry, the ratios of sides in right triangles form the basis of trigonometric functions.A proportion, on the other hand, states that two ratios are equal. In our exercise, the trigonometric functions such as sine, cosine, and tangent create proportions between the sides of a right triangle and help solve unknowns. For instance:

• \( \sin(\theta) = \frac{y}{r} \): This expresses the ratio of the length of the opposite side (y) to the hypotenuse (r).
• Rationalizing denominators involves maintaining proportions in simplified forms, such as converting \( \frac{2}{\sqrt{5}} \) to \( \frac{2\sqrt{5}}{5} \).

Understanding these ratios helps you to make sense of relationships within geometric shapes and extends into other areas such as physics, where force ratios, speed, and other dynamics are explored. Mastering these concepts will make learning more advanced math topics easier.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, integrates algebra and geometry to define and solve geometric problems through coordinate systems. Points are represented as coordinate pairs \((x, y)\), which are distances from the perpendicular lines of the x-axis and y-axis, respectively.In the context of trigonometric problems in the coordinate plane:

• The point \((-1, 2)\) helps identify the position of an angle's terminal side when theta is in standard position, showing direction from the origin.
• The distance from the origin (\(r\)) is calculated using the distance formula: \( r = \sqrt{x^2 + y^2} \).
• Once r is determined, it sets the stage for determining trigonometric function values by creating a right triangle with x and y as perpendicular components.

Through coordinate geometry, we connect algebraic equations and geometric representations. This gives a clearer tool for visualizing and solving problems not limited to our traditional number lines and shapes.