Problem 5 - Algebra and Trigonometry

Question

If \(P=(a, b)\) is a point on the terminal side of the angle \(\theta\) at a distance \(r\) from the origin, then \(\tan \theta=\) ________.

Step-by-Step Solution

Answer

tan(θ) = b/a

Step 1: Understand the problem

Identify the coordinates of the point P and the distance r. P is given as \(a, b\) and r is the distance from the origin to P.

Step 2: Recall the definition of tangent

The tangent of an angle \(\theta\) in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Step 3: Apply the definition to the given point

For the point \(P = (a, b)\), a is the x-coordinate (adjacent side) and b is the y-coordinate (opposite side). Therefore, \(\tan \theta = \frac{b}{a}\).

Step 4: Write the final expression for \(\tan \theta\)

The final expression for the tangent of the angle \(\theta\) when the point P is at \( (a, b) \) is \(\tan \theta = \frac{b}{a} \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios are a fundamental concept in trigonometry. They define the relationships between the angles and sides of a right triangle. There are three primary trigonometric ratios: sine, cosine, and tangent.
One way to remember these ratios is through the mnemonic SOH-CAH-TOA:

• Sine (sin) = Opposite / Hypotenuse
• Cosine (cos) = Adjacent / Hypotenuse
• Tangent (tan) = Opposite / Adjacent

The tangent function, in particular, is handy for finding the slope of an angle in a triangle. Understanding these ratios allows you to solve numerous problems related to angles and distances in various contexts, such as in right triangles and coordinate geometry.

Right Triangle

A right triangle is a type of triangle that includes one angle measuring exactly 90 degrees. This makes it incredibly useful for trigonometry due to the fixed relationship between its sides. Consider a right triangle with the right angle at vertex C, and the other angles at vertices A and B.
This triangle has three sides:

• The hypotenuse (the longest side opposite the right angle)
• The opposite side (opposite to the angle of interest)
• The adjacent side (next to the angle of interest)

Using right triangles, you can apply trigonometric ratios such as tangent to find unknown values. For instance, the tangent of an angle \( \theta \) is the ratio of the length of the opposite side to the length of the adjacent side, which simplifies problem-solving in geometric contexts.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using the coordinate plane to study geometric figures. It bridges algebra and geometry through graphs and equations. In coordinate geometry, you plot points, lines, and curves to solve for unknown values and understand spatial relationships.
For example, let's take the point \(P = (a, b)\) on the coordinate plane. In this case, 'a' and 'b' are the coordinates of the point 'P', and they represent the lengths of the adjacent and opposite sides, respectively, when considering the distance from the origin.
Given this setup in coordinate geometry, we can apply trigonometric concepts effectively. Particularly, to find the tangent of an angle \( \theta \) where the point \( P = (a, b) \) lies on the terminal side, we use:
\[ \tan \theta = \frac{b}{a} \]
This formula relates the coordinates of the point directly to the trigonometric function, making it very useful for solving problems that involve angles and distances on the coordinate plane.