Problem 4 - Technical Mathematics with Calculus

Question

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(7.93,8.27)$$

Step-by-Step Solution

Answer

Distance r is \(\sqrt{(7.93)^2 + (8.27)^2}\), six trig functions are defined with these coordinates, and the angle \(\theta\) is computed using \(\arctan(\frac{y}{x})\) or equivalent.

Step 1: Sketch the angle

Plot the point (7.93, 8.27) on the Cartesian plane. Draw the angle in standard position, which means that its initial side is on the positive x-axis. The terminal side of the angle passes through the given point.

Step 2: Compute the distance r

Calculate the distance r from the origin (0,0) to the point (7.93, 8.27) using the distance formula: \(r = \sqrt{x^2 + y^2}\). Substitute \(x = 7.93\) and \(y = 8.27\) and compute r.

Step 3: Write the six trigonometric functions

Use the values of x, y, and r to write the six trigonometric functions of the angle. Sine \(\sin(\theta) = \frac{y}{r}\), cosine \(\cos(\theta) = \frac{x}{r}\), tangent \(\tan(\theta) = \frac{y}{x}\), cosecant \(\csc(\theta) = \frac{1}{\sin(\theta)}\), secant \(\sec(\theta) = \frac{1}{\cos(\theta)}\), and cotangent \(\cot(\theta) = \frac{1}{\tan(\theta)}\).

Step 4: Find the angle \(\theta\)

Determine the angle \(\theta\) in radians or degrees using the inverse trigonometric functions. You can use either \(\arctan\), \(\arcsin\), or \(\arccos\), based on the x and y coordinates to find the angle in the correct quadrant.

Key Concepts

Sketching Angles

Understanding how to sketch angles is fundamental in trigonometry. An angle is formed by rotating a ray, known as the initial side, about its endpoint. In standard position, this initial side lies along the positive x-axis. The rotation determines the size and direction of the angle, and the ray that has been rotated is called the terminal side.

When sketching an angle in standard position, one must plot the terminal side as it passes through a given point on the Cartesian plane. For instance, with the point (7.93, 8.27), you would draw a line segment from the origin (0,0) to this point, which marks the terminal side. This visual representation helps to conceptualize the angle's measurement and sets the stage for finding trigonometric functions related to the angle.

Make the sketch clear and accurate. This provides a visual basis for further calculations, establishing a meaningful connection between the algebraic and geometric interpretations of the angle.

Distance Formula

The distance formula is a critical tool for finding the length of a segment connecting two points on the coordinate plane. It's derived from the Pythagorean theorem and is written as:
\[r = \sqrt{x^2 + y^2}\]
where \(r\) is the distance from the origin to a point with coordinates \((x, y)\). This formula is essential for determining the hypotenuse of a right triangle when the triangle's legs are along the x and y axes.

Using our example of point (7.93, 8.27), we plug the coordinates into the formula yielding \(r = \sqrt{(7.93)^2 + (8.27)^2}\), thus calculating the distance. This distance becomes particularly important as it's used in the definition of the six trigonometric functions and assists in solving real-world problems involving distance.

Standard Position of an Angle

Angles in trigonometry are most commonly expressed in standard position. This means the initial side of the angle lies on the positive x-axis, and the vertex is at the origin (0,0) of the coordinate system. The angle is positive if the rotation from the initial side to the terminal side is counter-clockwise and negative for a clockwise rotation.

When an angle is in standard position, its terminal side can be in any of the four quadrants. The quadrants are important because they determine the sign of the trigonometric functions for the angle. For example, sine and cosine are positive in the first quadrant but have different signs in other quadrants. Sketching an angle in standard position sets a clear context for these concepts and helps to determine the correct values for trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse processes of normal trigonometric functions. They are used to find an angle when the value of a trigonometric function is known. There are three main inverse functions:

• \(\arcsin(x)\) for the inverse sine,
• \(\arccos(x)\) for the inverse cosine, and
• \(\arctan(x)\) for the inverse tangent.

Each of these can be used to find an angle \(\theta\) given the ratio of sides in a right triangle. It's crucial to consider the quadrant in which the angle lies to ensure the correct angle measurement is determined. For instance, the tangent function can have the same value for two different angles in two different quadrants, so the context is key. Using our example, if we know the x and y coordinates of the point through which the terminal side passes, we could use \(\arctan(\frac{y}{x})\) to find \(\theta\). However, remember to adjust the angle based on the quadrant to get the correct angle in standard position.