Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and the terminal side of \(\theta\) is in the specified quadrant and satisfies the given condition. IV: on the line \(3 y+5 x=0\)

Step-by-Step Solution

Answer

Sine: \(-\frac{5}{\sqrt{34}}\), Cosine: \(\frac{3}{\sqrt{34}}\), Tangent: \(-\frac{5}{3}\), Cosecant: \(-\frac{\sqrt{34}}{5}\), Secant: \(\frac{\sqrt{34}}{3}\), Cotangent: \(-\frac{3}{5}\).

Step 1: Understand the Condition

The given line is \(3y + 5x = 0\). This can be rewritten in slope-intercept form \(y = -\frac{5}{3}x\). This line goes through the origin and has a negative slope, indicating it goes in Quadrants II and IV.

Step 2: Choose a Point in Quadrant IV

To find a point on the line that lies in Quadrant IV, let's choose \(x=3\). Substitute into the line equation: \(y = -\frac{5}{3}(3) = -5\). Thus, the point is \((3, -5)\). This point is indeed in Quadrant IV where x is positive and y is negative.

Step 3: Determine the Hypotenuse

Using the point \((3, -5)\) to form a right triangle with the x-axis, use the Pythagorean theorem to find the hypotenuse: \(r = \sqrt{x^2 + y^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}\).

Step 4: Calculate Sine and Cosine

For the point \((3, -5)\) with hypotenuse \(r = \sqrt{34}\), calculate the trigonometric functions:- \(\sin(\theta) = \frac{y}{r} = \frac{-5}{\sqrt{34}}\)- \(\cos(\theta) = \frac{x}{r} = \frac{3}{\sqrt{34}}\).

Step 5: Calculate Other Trigonometric Functions

Now find the other four trigonometric functions:- \(\tan(\theta) = \frac{y}{x} = \frac{-5}{3}\)- \(\csc(\theta) = \frac{1}{\sin(\theta)} = -\frac{\sqrt{34}}{5}\)- \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{34}}{3}\)- \(\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{3}{5}\).

Key Concepts

Understanding Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four distinct sections known as quadrants. These quadrants help us determine the sign of trigonometric functions:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

The angle \( \theta \) in our exercise is in Quadrant IV, indicating the point on the coordinate plane associated with \( \theta \) has a positive x-value and a negative y-value. This is essential when evaluating the signs of trigonometric ratios because these ratios involve the coordinates of a point in the quadrant.

Applying the Pythagorean Theorem

The Pythagorean theorem is a crucial tool in finding the hypotenuse in a right triangle. Given a point \( (x, y) \), the theorem states:\[ r = \sqrt{x^2 + y^2} \]where \( r \) is the hypotenuse. This relationship is foundational when working with right triangles to find unknown sides. In the problem, the point (3, -5) provided the x and y values, and the Pythagorean theorem helped calculate the hypotenuse as \( \sqrt{34} \).
Knowing \( r \) as the distance from the origin to the point on the terminal side of \( \theta \) helps in determining the trigonometric functions.

The Standard Position of Angles

In the study of trigonometry, angles are measured from a standard starting point called the 'standard position.' An angle is in standard position when its vertex is at the origin of the coordinate plane:

• The initial side is along the positive x-axis.
• The terminal side is where the angle ends after the rotation from the x-axis.

This concept is important as it allows us to utilize the coordinate plane to define trigonometric functions according to the position of the terminal side in various quadrants. Understanding standard position helps students interpret and solve problems, such as identifying the quadrant where an angle is located, as seen in the exercise.

Exploring Right Triangles in Trigonometry

Right triangles naturally occur in trigonometry, where one angle is always 90 degrees. This configuration makes it possible to define sine, cosine, tangent, and other trigonometric functions based on ratios of the triangle's sides.

• The opposite side is the side directly opposite the angle in question.
• The adjacent side lies next to the angle in question but is not the hypotenuse.
• The hypotenuse is the longest side, opposite the right angle.

In our scenario, the triangle is formed by the line segment from the origin to the point (3, -5), creating a right triangle with legs 3 and -5, with \( \sqrt{34} \) as the hypotenuse. This setup simplified the calculation of trigonometric functions, forming the basis for determining exact values.