Problem 19 - 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. $$(7,24)$$

Step-by-Step Solution

Answer

The six trigonometric values for the given point are: \( sin = \frac{24}{25} \), \( cos = \frac{7}{25} \), \( tan = \frac{24}{7} \), \( csc = \frac{25}{24} \), \( sec = \frac{25}{7} \), \( cot = \frac{7}{24} \).

Step 1: Calculate the Radius (r)

Calculate the radius using the Pythagorean theorem for the right triangle formed by the x (7) and y (24) coordinates. The radius \( r \) is given by \( r = \sqrt{x^2 + y^2} \). Therefore, \( r = \sqrt{7^2 + 24^2} = 25 \).

Step 2: Calculate Sine (sin), Cosine (cos) and Tangent (tan)

The values for sin, cos, and tan can be found using the definitions of these ratios in a right triangle. \( sin = \frac{y}{r} \), \( cos = \frac{x}{r} \), and \( tan = \frac{y}{x} \). Therefore, \( sin = \frac{24}{25} \), \( cos = \frac{7}{25} \), and \( tan = \frac{24}{7} \).

Step 3: Calculate Cosecant (csc), Secant (sec) and Cotangent (cot)

These ratios are the reciprocal of sin, cos, and tan. Therefore, \( csc = \frac{1}{sin} \), \( sec = \frac{1}{cos} \), and \( cot = \frac{1}{tan} \). Hence, \( csc = \frac{25}{24} \), \( sec = \frac{25}{7}, and \( cot = \frac{7}{24}.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry. It expresses a special relationship between the sides of a right triangle. In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it is written as:\[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
This theorem is essential for finding the length of a side when two other sides are known. When solving problems with coordinates, like the one given in the original exercise, you utilize the Pythagorean theorem to find the radius of the circle that the triangle fits into. In this context, the radius functions as the hypotenuse. For our specific point \((7, 24)\), the radius \( r \) is calculated as:\[ r = \sqrt{7^2 + 24^2} = 25 \].
This calculation is crucial for determining the trigonometric functions for an angle.

right triangle

In geometry, a right triangle is a type of triangle that has one angle measuring 90 degrees. This distinctive right angle creates unique properties and formulas like the Pythagorean theorem. Understanding how to work with a right triangle helps us utilize trigonometric functions. These functions include sine, cosine, and tangent, which are key in solving a variety of mathematical problems.
A right triangle consists of:

• One right angle (90 degrees)
• Two acute angles (less than 90 degrees)
• Three sides: hypotenuse (opposite the right angle), and two legs (one adjacent to the right angle)

When dealing with trigonometric functions, you usually calculate ratios based on the sides of a right triangle. For example, given point \((7, 24)\), these sides form legs of the right triangle with the hypotenuse being \( 25 \) as calculated using the Pythagorean theorem.

reciprocal trigonometric functions

Reciprocal trigonometric functions are the inverses of the basic trigonometric functions. These functions provide additional ways to explore the relationships between the sides and angles of a right triangle.
Here are the three reciprocal functions:

• Cosecant (\(csc\)): The reciprocal of sine. \(csc(\theta) = \frac{1}{sin(\theta)}\)
• Secant (\(sec\)): The reciprocal of cosine. \(sec(\theta) = \frac{1}{cos(\theta)}\)
• Cotangent (\(cot\)): The reciprocal of tangent. \(cot(\theta) = \frac{1}{tan(\theta)}\)

With the original problem, the point \((7, 24)\) gives us the angle whose reciprocal functions can be calculated as follows:

• \(csc = \frac{25}{24}\)
• \(sec = \frac{25}{7}\)
• \(cot = \frac{7}{24}\)

This approach helps in finding more solutions to trigonometric problems where only angles are given.

standard position of an angle

In trigonometry, the standard position of an angle is a way of visualizing angles and their relationships with coordinates. When an angle is in standard position, its vertex is at the origin of the coordinate plane, with one arm lying along the positive x-axis.
This setup is useful because it provides a consistent method for describing the location and measurements of angles. The terminal side of the angle, which is the other arm, swings through the quadrants as the angle increases.

• Positive angles are measured counterclockwise from the positive x-axis.
• Negative angles are measured clockwise.

In our original exercise's point \((7, 24)\), it lies on the terminal side of the angle in standard position. By using the coordinates, we leverage both Cartesian plane geometry and trigonometric principles to find exact values for trigonometric functions of that angle. Understanding standard position is vital for correctly interpreting and solving problems involving angles on the coordinate plane.