Problem 18 - Precalculus: A Concise Course

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. $$ \left(3 \frac{1}{2},-7 \frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions of the angle are \(\sin(\theta) = -\frac{31}{2\sqrt{1081}}\), \(\cos(\theta) = \frac{7}{\sqrt{1081}}\), \(tan(\theta) = -\frac{62}{7}\), \(csc(\theta) = \frac{2\sqrt{1081}}{31}\), \(sec(\theta) = \frac{\sqrt{1081}}{7}\) and \(cot(\theta) = -\frac{7}{62}\)

Step 1: Conversion of Coordinates into Fractions

Convert the mixed numbers into improper fractions. Therefore, the coordinates become: \((\frac{7}{2},-\frac{31}{4})\)

Step 2: Calculating the Radius/Hypotenuse

Calculate the radius (r), which is the hypotenuse of the right triangle. Use the Pythagorean Theorem: \(r = \sqrt{(x^2 + y^2)}\). Therefore, \(r = \sqrt{(\frac{7}{2})^2 + (-\frac{31}{4})^2} = \sqrt{\frac{49}{4} + \frac{961}{16}} = \sqrt{\frac{196}{4} + \frac{1924}{4}} = \sqrt{\frac{1081}{2}}\)

Step 3: Determining Trigonometric Function Values

Determine the exact values of the six trigonometric functions as follows: \(\sin(\theta) = \frac{y}{r} = -\frac{31}{4} \div \sqrt{\frac{1081}{2}} = -\frac{31}{2\sqrt{1081}}\), \(\cos(\theta) = \frac{x}{r} = \frac{7}{2} \div \sqrt{\frac{1081}{2}} = \frac{7}{\sqrt{1081}}\), \(tan(\theta) = \frac{y}{x} = -\frac{31}{4} \div \frac{7}{2} = -\frac{62}{7}\), \(csc(\theta) = \frac{1}{\sin(\theta)} = \frac{2\sqrt{1081}}{31}\), \(sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{1081}}{7}\), and \(cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{7}{62}\).

Key Concepts

Pythagorean Theorem

Understanding the Pythagorean Theorem is crucial when dealing with right triangles, and it plays a fundamental role in trigonometry.
The theorem states that in a right-angled 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 relationship is commonly written as
\[a^2 + b^2 = c^2\].
When applied to our problem, if we treat the x and y coordinates of the point as the lengths of the two legs of a right triangle, we can find the hypotenuse, which in trigonometry is often referred to as the radius (r), with the origin as the triangle's right angle. Calculating the hypotenuse is a crucial step in determining the values of trigonometric functions as it represents the distance from the origin to the point on the coordinate plane.

Standard Position Angle

In trigonometry, an angle in standard position is an angle with its vertex at the origin of the coordinate plane and its initial side along the positive x-axis.
The terminal side of the angle is the position of the line after the angle has been rotated from its initial position. The angle is measured in counter-clockwise direction from the initial side to the terminal side. An angle in standard position can be identified to be in different quadrants based on where the terminal side lies, impacting the signs of the trigonometric functions. For example, if the point lies in the second quadrant, the x-coordinate is negative while the y-coordinate is positive, resulting in a positive sine value and a negative cosine value.

Exact Values Trigonometry

When finding the exact values of trigonometric functions, we want to avoid approximations and use the most precise form possible, often involving square roots and fractions.
To find these values, we use the coordinates of the point along with the hypotenuse. For the sine function, we divide the y-coordinate by the hypotenuse, for the cosine function, we use the x-coordinate, and for the tangent, we divide y by x.
Because we are finding exact values, we leave our answers in terms of square roots and fractions rather than decimal approximations. This precise form is crucial in higher mathematics and enables one to better understand the properties and relationships between different trigonometric functions.