Problem 4 - Contemporary Precalculus

Question

Evaluate the trigonometric functions at the angle (in standard position) whose terminal side contains the given point. $$(\sqrt{2}, \sqrt{3})$$

Step-by-Step Solution

Answer

Question: Find the cosine, sine, and tangent of an angle in standard position whose terminal side contains the point \((\sqrt{2}, \sqrt{3})\). Answer: The trigonometric functions for the given angle are: \(\cos(\theta) = \frac{\sqrt{2}}{\sqrt{5}}\), \(\sin(\theta) = \frac{\sqrt{3}}{\sqrt{5}}\), and \(\tan(\theta) = \frac{\sqrt{3}}{\sqrt{2}}\).

Step 1: Find the Reference Angle

Using the given coordinates \((\sqrt{2}, \sqrt{3})\), we can find the reference angle by using the Pythagorean theorem as follows: $$ r = \sqrt{(x^2 + y^2)} $$ We have \(x = \sqrt{2}\) and \(y = \sqrt{3}\), so: $$ r = \sqrt{((\sqrt{2})^2 + (\sqrt{3})^2)} $$ Which simplifies to: $$ r = \sqrt{(2 + 3)} = \sqrt{5} $$ Now that we know the hypotenuse, we can find the trigonometric functions.

Step 2: Find Cosine

The cosine of the angle can be found with the given formula: $$ \cos(\theta) = \frac{x}{r} $$ Using the values we found earlier, we get: $$ \cos(\theta) = \frac{\sqrt{2}}{\sqrt{5}} $$

Step 3: Find Sine

The sine of the angle can be found with the given formula: $$ \sin(\theta) = \frac{y}{r} $$ Using the values we found earlier, we get: $$ \sin(\theta) = \frac{\sqrt{3}}{\sqrt{5}} $$

Step 4: Find Tangent

The tangent of the angle can be found with the given formula: $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$ Using the values we found earlier, we get: $$ \tan(\theta) = \frac{\frac{\sqrt{3}}{\sqrt{5}}}{\frac{\sqrt{2}}{\sqrt{5}}} = \frac{\sqrt{3}}{\sqrt{2}} $$ Thus, the trigonometric functions for the given angle are: $$ \cos(\theta) = \frac{\sqrt{2}}{\sqrt{5}}, \sin(\theta) = \frac{\sqrt{3}}{\sqrt{5}}, \text{ and } \tan(\theta) = \frac{\sqrt{3}}{\sqrt{2}} $$

Key Concepts

Reference Angle

A reference angle is a useful concept in trigonometry that helps us find trigonometric values for any angle, even when it’s not one of the angle measures we commonly use, such as 0, 30, 45, 60, 90 degrees, etc.
Given any angle, its reference angle is the closest angle it makes with the x-axis. It's always a positive angle and less than or equal to 90 degrees.
In our problem, we use the point \(\sqrt{2}, \sqrt{3}\) on the coordinate plane to find the length of the hypotenuse with the Pythagorean theorem, which acts as a precursor to understanding the reference angle.
Calculating \(r = \sqrt{(x^2 + y^2)}\) provides the necessary foundational knowledge of the triangle formed, leading us to a better understanding of the primary position of our angle relative to the Cartesian plane.

Cosine

The cosine function, \(\cos(\theta)\), is one of the fundamental trigonometric functions that relates the adjacent side of a right triangle to its hypotenuse.
For our specific case, the adjacent side corresponds to the x-coordinate \(x = \sqrt{2}\), and the hypotenuse is \(r = \sqrt{5}\).
The formula \(\cos(\theta) = \frac{x}{r}\) gives us the cosine of the angle as \(\frac{\sqrt{2}}{\sqrt{5}}\).
Understanding this concept is essential when dealing with angles found in standard position since it allows for direct application in coordinate systems and helps calculate other trigonometric functions.

Sine

The sine function, \(\sin(\theta)\), represents the ratio of the opposite side to the hypotenuse in a right triangle.
In our example, the opposite side is the y-coordinate, given as \(y = \sqrt{3}\), and the hypotenuse is \(r = \sqrt{5}\).
The calculation \(\frac{y}{r}\) allows us to find \(\sin(\theta)\) as \(\frac{\sqrt{3}}{\sqrt{5}}\), indicating how the angle in standard position projects onto the y-axis.
The sine function is crucial for understanding vertical components of angles when analyzing coordinate points, providing insight into the shape and orientation of triangles relative to the x-axis.

Tangent

The tangent function, often written as \(\tan(\theta)\), is the ratio of the sine to the cosine of a given angle.
It is profoundly helpful for finding inclinations or slopes that result from the angle in question.
In this context, \(\tan(\theta)\) combines previous calculations: \(\frac{\sin(\theta)}{\cos(\theta)}\).
The resulting value, \(\frac{\sqrt{3}}{\sqrt{2}}\), demonstrates how this function synthesizes information from sine and cosine, forming a numeric expression of the angle's steepness relating to x and y axes in their respective quadrants.
Conceptually, the tangent function completes our understanding of how angles transform in trigonometric terms, encapsulating horizontal and vertical relations between the coordinates.