Problem 106 - Precalculus

Question

If the terminal side of angle \(\theta\) passes through the point \((-3 a, 4 a),\) find \(\sin \theta .\) Assume \(a>0\)

Step-by-Step Solution

Answer

\(\sin \theta = \frac{4}{5}\).

Step 1: Understand the problem

We need to find \( \sin \theta \) where the terminal side of the angle \( \theta \) passes through the point \((-3a, 4a)\). We are given that \(a > 0\). This point gives us the coordinates \(x = -3a\) and \(y = 4a\). The sine of an angle in a right triangle is the ratio of the opposite side over the hypotenuse.

Step 2: Calculate the hypotenuse using Pythagorean theorem

The hypotenuse can be found by applying the Pythagorean theorem to the right triangle formed. The formula is \( r = \sqrt{x^2 + y^2} \). Substituting in the given values, we find \[ r = \sqrt{(-3a)^2 + (4a)^2} = \sqrt{9a^2 + 16a^2} = \sqrt{25a^2} = 5a. \]

Step 3: Find \( \sin \theta \)

The sine of angle \( \theta \) is \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \). From the point \((-3a, 4a)\), the opposite side is the \(y\)-coordinate \(4a\). Therefore, \( \sin \theta = \frac{4a}{5a} = \frac{4}{5}. \) The \(a\)'s cancel out as they are positive and contribute equally to both numerator and denominator.

Key Concepts

Sine function

The sine function is a fundamental concept in trigonometry, providing a way to relate angles of a triangle to the ratios of its sides. Specifically, in a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse. To express this using the angle \( \theta \): \[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \].

This function is crucial for analyzing relationships within right triangles, particularly in scenarios involving angles and side lengths. It transforms direct physical quantities (lengths of segments) into a dimensionless number through the trigonometric identity, allowing for easier computations. In coordinate geometry, when dealing with points, like \((-3a, 4a)\), the "opposite side" becomes the \(y\)-coordinate (4a), while the hypotenuse is derived from the distance formula formed by an origin point \((0,0)\) to the point \((-3a, 4a)\). This makes sine useful in bridging geometry with algebra.

Pythagorean theorem

The Pythagorean theorem is a principle in geometry that connects the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse, then: \[ c^2 = a^2 + b^2 \].

In coordinate geometry, this theorem helps to determine distances between points across a plane. For example, consider the exercise where

• \(x = -3a\) and \(y = 4a\),
• The hypotenuse, \(r\), is calculated using: \[ r = \sqrt{(-3a)^2 + (4a)^2} \],
• This simplifies to \(r = 5a\).

By using this formula, one can easily compute the hypotenuse when dealing with angles and points in a plane.

This theorem is foundational in connecting geometrical perspectives to trigonometric functions, essential for solving complex problems like determining sine from coordinate points.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric figures within a numerical framework. This branch of geometry uses a coordinate system to define and analyze geometrical shapes, providing a bridge between algebra and geometry.

• Points are defined by coordinates \((x, y)\) in a plane.
• Distance and lengths connecting these points can be computed.

In our scenario, the point \((-3a, 4a)\) lies on the terminal side of the angle \(\theta\). This representation helps us determine the lengths of the sides of a corresponding right triangle by using those coordinates. Through such calculations:

• The horizontal change is the difference in the \(x\)-coordinates, here \(-3a\).
• The vertical change is captured by the \(y\)-coordinate, here \(4a\).
• The hypotenuse or the resultant distance is determined using the Pythagorean theorem.

Coordinate geometry makes the solutions to trigonometric problems more intuitive by visually connecting algebraic calculations with geometrical representations.