Question

Challenge Problem If the terminal side of an angle contains the point \((5 n,-12 n)\) with \(n>0,\) find \(\sin \theta\)

Short answer

\(\sin \theta = -\frac{12}{13}\)

Step-by-step solution

- Understand the coordinates

The point \( (5n, -12n) \) lies on the terminal side of the angle \( \theta \). This means the x-coordinate is \( 5n \) and the y-coordinate is \( -12n \).

- Find the radius

The radius (or hypotenuse) can be found using the Pythagorean theorem: \[ r = \sqrt{(5n)^2 + (-12n)^2} \].

- Simplify and compute the radius

Calculate \[ r = \sqrt{25n^2 + 144n^2} = \sqrt{169n^2} = 13n \].

- Define the sine function

The sine of an angle is defined as the ratio of the opposite side to the hypotenuse: \[ \sin \theta = \frac{y}{r} = \frac{-12n}{13n} \].

- Simplify the sine ratio

Simplify the ratio to find \[ \sin \theta = \frac{-12n}{13n} = -\frac{12}{13} \].

Key concepts

sine function

The sine function is one of the primary functions in trigonometry. It relates to the ratio of the length of the side opposite a given angle in a right triangle to the hypotenuse.
For an angle \(\theta\) in the coordinate plane, the sine function can be defined using the coordinates of a point \( (x, y) \) on the terminal side of the angle and the radius (hypotenuse) \( r \).
In our exercise, the sine function is calculated as:
\[ \sin \theta = \frac{y}{r} \]
Here, \( y \) is the y-coordinate and \( r \) is the hypotenuse. Using the given coordinates \( (5n, -12n) \), we find that:
\[ \sin \theta = \frac{-12n}{13n} = -\frac{12}{13} \]
This means the sine of the angle is \(-\frac{12}{13}\).
Understanding sine is crucial for solving many trigonometric problems.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The formula is:
\[ a^2 + b^2 = c^2 \]
Where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the hypotenuse. In our problem, the coordinates are \( 5n \) and \(-12n \). We use the theorem to find the hypotenuse \( r \):
\[ r = \sqrt{(5n)^2 + (-12n)^2} \]
Calculating this, we get:

• \[ (5n)^2 = 25n^2 \]
• \[ (-12n)^2 = 144n^2 \]
• \[ 25n^2 + 144n^2 = 169n^2 \]

Thus, \( r = \sqrt{169n^2} = 13n \).
This shows how the Pythagorean theorem helps determine the hypotenuse.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system.
It allows us to describe geometric shapes numerically and derive relationships using algebra. In the problem, the coordinates \( (5n, -12n) \) lie on the terminal side of the angle \(\theta\).
Key points to remember:

• Coordinates represent points on the plane.
• The x-coordinate is horizontal distance.
• The y-coordinate is vertical distance.
• The distance from the origin (0,0) to the point \((x, y)\) is the radius.

For the given point, the x-coordinate is \( 5n \) and the y-coordinate is \(-12n \). Using coordinate geometry helps in applying the Pythagorean theorem to find the hypotenuse and then using it to determine trigonometric functions.

terminal side of an angle

The terminal side of an angle is the position of the angle measured in standard position, starting from the positive x-axis and rotating to the angle's terminal side.
Understanding the terminal side is crucial in trigonometry as it defines the coordinates used in calculations.
When an angle is in standard position:

• Its vertex is at the origin \( (0, 0) \).
• Its initial side lies on the positive x-axis.
• The terminal side is where the angle measure ends.

In our problem, the terminal side of angle \(\theta \) includes the point \((5n, -12n)\). From this point, we can use trigonometric functions to solve for the angle's properties.
By knowing the coordinates and the hypotenuse, we calculate the sine of the angle using the formula \( \sin \theta = \frac{y}{r} \).
Terminal angles help to determine precise trigonometric values using coordinate geometry.