Problem 20 - 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. $$(-4,-3)$$

Step-by-Step Solution

Answer

The trigonometric functions are: \(\sin(\theta) = -\frac{3}{5}\), \(\cos(\theta) = -\frac{4}{5}\), \(\tan(\theta) = \frac{3}{4}\), \(\csc(\theta) = -\frac{5}{3}\), \(\sec(\theta) = -\frac{5}{4}\), \(\cot(\theta) = \frac{4}{3}\).

Step 1: Determine the Quadrant

The point \((-4, -3)\) is located in the third quadrant of the coordinate plane because both the x-coordinate and the y-coordinate are negative.

Step 2: Calculate the Distance from Origin

Use the distance formula to find the length of the hypotenuse: \(\text{distance} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).This distance represents the hypotenuse (\(r\)) of the right triangle formed by the point.

Step 3: Find the Sine

The sine of the angle \(\theta\) is the ratio of the opposite side to the hypotenuse: \[ \sin(\theta) = \frac{-3}{5} \].

Step 4: Find the Cosine

The cosine of the angle \(\theta\) is the ratio of the adjacent side to the hypotenuse: \[ \cos(\theta) = \frac{-4}{5} \].

Step 5: Find the Tangent

The tangent of the angle \(\theta\) is the ratio of the opposite side to the adjacent side: \[ \tan(\theta) = \frac{-3}{-4} = \frac{3}{4} \].

Step 6: Find the Cosecant

The cosecant is the reciprocal of the sine:\[ \csc(\theta) = \frac{5}{-3} = -\frac{5}{3} \].

Step 7: Find the Secant

The secant is the reciprocal of the cosine: \[ \sec(\theta) = \frac{5}{-4} = -\frac{5}{4} \].

Step 8: Find the Cotangent

The cotangent is the reciprocal of the tangent: \[ \cot(\theta) = \frac{4}{3} \].

Key Concepts

Standard Position

In trigonometry, an angle is said to be in standard position when its vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis. This is a common way to present angles in order to simplify the evaluation of trigonometric functions such as sine, cosine, and tangent. To sketch an angle in standard position, you start from the positive x-axis and move in the counterclockwise direction unless the angle is negative, in which case the movement is clockwise.

• The initial side remains stable along the x-axis.
• The terminal side is where the angle ends.

Considering the point \((-4, -3)\) from the original exercise, this point lies on the terminal side of an angle in the third quadrant of the coordinate plane. Therefore, the angle in standard position would swing from the positive x-axis to that point, following the negative direction on both axes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the use of coordinates to describe geometric figures and the relationships between them. When working with angles and trigonometric functions, we often use points on the coordinate plane to understand the position and nature of these angles.

• Each point is identified by an ordered pair \((x, y)\).
• These coordinates can be used to form segments and triangles which help in calculating trigonometric ratios.

For a given point like \((-4, -3)\), these coordinates tell us exactly where the point is located on the plane. By understanding its location concerning the origin, we can visualize the right triangle created by drawing perpendiculars to the axes and use this triangle for trigonometric calculations.

Right Triangle Trigonometry

Right triangle trigonometry is foundational for understanding and calculating the trigonometric values of angles. It applies the Pythagorean theorem and definitions of trigonometric functions based on the sides of a right triangle.To find trigonometric values like sine, cosine, and tangent, we use:

• The opposite side: The side opposite the angle in question.
• The adjacent side: The side next to the angle, not counting the hypotenuse.
• The hypotenuse: The longest side of the right triangle, opposite the right angle.

In our exercise, the point \((-4, -3)\) forms a right triangle with:- \(r = 5\) as the hypotenuse,- \(x = -4\) the base or adjacent, and- \(y = -3\) the height or opposite.Using these, we find:- \(\sin(\theta) = \frac{-3}{5}\)- \(\cos(\theta) = \frac{-4}{5}\)- \(\tan(\theta) = \frac{3}{4}\)This demonstrates how the point on the coordinate plane helps in deriving these ratios.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four sections, known as quadrants, numbered counter-clockwise starting from the upper right. Understanding these quadrants is essential to working with trigonometric functions since the sign of these functions can change based on the quadrant in which the angle is located.

• First Quadrant: Both x and y are positive; all trigonometric functions are positive.
• Second Quadrant: x is negative, y is positive; only sine and cosecant are positive.
• Third Quadrant: Both x and y are negative; only tangent and cotangent are positive.
• Fourth Quadrant: x is positive, y is negative; only cosine and secant are positive.

Given the point \((-4, -3)\), which falls in the third quadrant, the signs reveal much about our trigonometric functions:- Sine and cosine are negative,- Tangent is positive.This understanding helps predict and verify the expected values of trigonometric functions before performing detailed calculations.