Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(3,-9)$$

Step-by-Step Solution

Answer

\(\sin(\theta) = -\sqrt{10}/3, \cos(\theta) = \sqrt{10}/10, \tan(\theta) = -3, \csc(\theta) = -\sqrt{10}/9, \sec(\theta) = \sqrt{10}/3, \cot(\theta) = -1/3\)

Step 1: Identify the coordinates

The given point is (3,-9). This means that the coordinates of the point are \(x = 3\) and \(y = -9\). \(x\) and \(y\) represent, respectively, the length of the adjacent side and the opposite side in the right triangle formed within the circle inscribed in the Cartesian plane.

Step 2: Calculate the length of the hypotenuse (r)

Using the Pythagorean theorem, calculate the length of the hypotenuse (r = \(\sqrt{x^2 + y^2}\)). This gives \(r = \sqrt{(3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90}\).

Step 3: Calculate the values of the six trigonometric functions

The trigonometric functions are defined as follows: \(\sin(\theta) = y/r, \cos(\theta) = x/r, \tan(\theta) = y/x, \csc(\theta) = r/y, \sec(\theta) = r/x\), and \(\cot(\theta) = x/y\). Substituting \(x = 3\), \(y = -9\), and \(r = \sqrt{90}\) into these definitions gives: \(\sin(\theta) = -9/\sqrt{90} = -\sqrt{10}/3, \cos(\theta) = 3/\sqrt{90} = \sqrt{10}/10, \tan(\theta) = -9/3 = -3, \csc(\theta) = \sqrt{90}/-9 = -\sqrt{10}/9, \sec(\theta) = \sqrt{90}/3 = \sqrt{10}/3\), and \(\cot(\theta) = 3/-9 = -1/3\)

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that describes the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle), is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the equation: \[ a^2 + b^2 = c^2 \]Where:

• \(a\) and \(b\) are the lengths of the legs (adjacent and opposite sides).
• \(c\) is the length of the hypotenuse.

In the context of the Cartesian plane, the theorem helps in determining the distance from the origin to a point \((x, y)\). By treating \(x\) and \(y\) as the legs of a right triangle, and the hypotenuse as the distance from the origin to the point \((x, y)\), the formula becomes:\[ r = \sqrt{x^2 + y^2} \]This is crucial for calculating properties like trigonometric functions, as these depend on the ratios of these triangle side lengths.

Cartesian plane

The Cartesian plane is a two-dimensional plane defined by a horizontal line (x-axis) and a vertical line (y-axis). Together, they form a grid that allows us to pinpoint locations using coordinates. Each point on this plane is represented by a pair of numbers \((x, y)\), which measures horizontal (x) and vertical (y) distances from the origin (0, 0). Here's what's important:

• The x-axis increases as you move to the right and decreases as you move to the left.
• The y-axis increases as you move upward and decreases as you move downward.
• Coordinates in the Cartesian plane are typically expressed as \((x, y)\).

The Cartesian plane is used extensively in mathematics, especially in trigonometry and algebra, to graph functions, solve equations, and conduct geometric transformations. When graphing a standard position angle, the Cartesian plane helps visualize the angle's terminal side starting from the origin.

Standard position angle

A standard position angle is a special way of positioning an angle in the coordinate plane. Its vertex is always at the origin \((0,0)\), and the initial side of the angle coincides with the positive x-axis. When determining trigonometric functions, the terminal side's location (the side opposite the initial side) in the plane is crucial.Here are the general characteristics:

• The angle is measured from the initial side counterclockwise to the terminal side, unless specified.
• Positive angles extend counterclockwise, while negative angles extend clockwise.
• Points on the terminal side can be associated with coordinates \((x, y)\), which represent real numbers.

With standard position angles, you have the benefit of utilizing the Cartesian plane to derive the six trigonometric functions by applying the point on the terminal side. By using the coordinate \((x, y)\) and the hypotenuse \(r\), these functions—sine, cosine, tangent, cosecant, secant, and cotangent—are easily found.