Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(\sqrt{2}, \sqrt{2})$$

Step-by-Step Solution

Answer

\(\theta = 45^\circ \); the trigonometric functions are \( \sin(\theta) = \frac{\sqrt{2}}{2}, \cos(\theta) = \frac{\sqrt{2}}{2}, \tan(\theta) = 1, \csc(\theta) = \sqrt{2}, \sec(\theta) = \sqrt{2}, \cot(\theta) = 1 \).

Step 1: Understanding the Problem

We need to sketch an angle \( \theta \) in standard position that passes through the point \((\sqrt{2}, \sqrt{2})\). Here, \( \theta \) is the angle between the x-axis and the line connecting the origin to the point.

Step 2: Identifying the Angle

The point \((\sqrt{2}, \sqrt{2})\) is in the first quadrant. In the first quadrant, the x and y coordinates are equal, indicating an angle of \(45^\circ\) or \(\frac{\pi}{4}\) radians from the x-axis.

Step 3: Calculating the Hypotenuse

To find the hypotenuse of the right triangle formed, we use the Pythagorean theorem: \[ r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = 2 \] where \( r \) is the hypotenuse.

Step 4: Calculating the Sine and Cosine

The sine of the angle is the ratio of the opposite side (y) to the hypotenuse \(r\):\[ \sin(\theta) = \frac{\sqrt{2}}{2} \]The cosine of the angle is the ratio of the adjacent side (x) to the hypotenuse \(r\):\[ \cos(\theta) = \frac{\sqrt{2}}{2} \]

Step 5: Calculating the Tangent

The tangent of the angle is the ratio of the opposite side (y) to the adjacent side (x):\[ \tan(\theta) = \frac{\sqrt{2}}{\sqrt{2}} = 1 \]

Step 6: Calculating Cosecant, Secant, and Cotangent

The cosecant is the reciprocal of sine:\[ \csc(\theta) = \frac{2}{\sqrt{2}} = \sqrt{2} \]The secant is the reciprocal of cosine:\[ \sec(\theta) = \frac{2}{\sqrt{2}} = \sqrt{2} \]The cotangent is the reciprocal of tangent:\[ \cot(\theta) = \frac{1}{1} = 1 \]

Step 7: Assembling the Trigonometric Functions

Summarizing, the trigonometric functions are:- \( \sin(\theta) = \frac{\sqrt{2}}{2} \)- \( \cos(\theta) = \frac{\sqrt{2}}{2} \)- \( \tan(\theta) = 1 \)- \( \csc(\theta) = \sqrt{2} \)- \( \sec(\theta) = \sqrt{2} \)- \( \cot(\theta) = 1 \)

Key Concepts

Standard Position

When we talk about an angle being in standard position, we are referring to a specific way we set up the angle in the coordinate plane. In standard position, the angle is

• Placed so that its vertex is at the origin, where the x-axis and y-axis intersect.
• The initial side of the angle lies along the positive x-axis.

This setup allows us to measure the angle from the x-axis, rotating counter-clockwise. If the terminal side of the angle passes through a specific point, like \(\sqrt{2}, \sqrt{2}\), this helps us identify specific trigonometric properties and values. Standard position is crucial for consistency across calculations in trigonometry. Understanding this concept helps us find the relationship between angles and their trigonometric functions.
It also provides a standardized way to deal with angles, ensuring everyone can visualize them in the same manner.

First Quadrant

The coordinate plane is divided into four quadrants, and when a point such as \(\sqrt{2}, \sqrt{2}\) is mentioned, the first quadrant becomes relevant. The first quadrant is

• Located in the upper right section of the coordinate plane, where both x and y coordinates are positive.
• A place where angles typically range from 0 to 90 degrees, or 0 to \frac{\pi}{2}\ radians.

When you sketch an angle in standard position, positioning that angle in the first quadrant provides clarity, as trigonometric functions like sine, cosine, and tangent all have positive values here. This is due to both the x and y-components being positive. By knowing you're in the first quadrant, you're acknowledging the positive characteristics of trigonometric ratios, which simplifies understanding and calculating these functions.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in trigonometry and geometry. It relates to right triangles and helps us calculate missing lengths when two sides are known. The theorem states that for a right triangle with legs a and b, and hypotenuse c, the equation is: \[ a^2 + b^2 = c^2 \]
In the context of trigonometric functions and the exercise provided, the

• Legs are the x and y coordinates of the point, \(\sqrt{2}\) and \(\sqrt{2}\) respectively.
• Hypotenuse is represented by the variable \(r\), which is calculated using these coordinates.

By plugging these values into the Pythagorean theorem, you determine the hypotenuse: \[ r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = 2 \]
This length is then used as a foundation for finding the different trigonometric ratios. This theorem is essential as it offers a direct relationship between the sides of a triangle, forming a basis for understanding angles and distances within the coordinate plane.