Avoiding common sign errors when calculating trig functions with x, y, r

📚 Understanding Trigonometric Functions with x, y, and r

Trigonometric functions relate angles to the ratios of sides of a right triangle. When working with a coordinate plane, we often define these functions in terms of a point (x, y) and its distance from the origin, r. Getting the signs right is crucial for accuracy.

📜 Background and Definitions

Historically, trigonometry was developed to study the relationships between angles and sides of triangles. The definitions of trigonometric functions in terms of x, y, and r extend this to all angles, not just those in a right triangle.

Let's define our terms:

• 📍 x: The horizontal coordinate of a point on the coordinate plane.
• 📈 y: The vertical coordinate of a point on the coordinate plane.
• 📏 r: The distance from the origin (0,0) to the point (x, y), calculated as $r = \sqrt{x^2 + y^2}$. Note that r is always non-negative.

Now, let's define the trigonometric functions:

• sin$\theta = \frac{y}{r}$
• cos$\theta = \frac{x}{r}$
• tan$\theta = \frac{y}{x}$
• csc$\theta = \frac{r}{y}$
• sec$\theta = \frac{r}{x}$
• cot$\theta = \frac{x}{y}$

🧭 Key Principles for Avoiding Sign Errors

• ➕ Quadrant I (x > 0, y > 0): All trigonometric functions are positive in the first quadrant.
• ➖ Quadrant II (x < 0, y > 0): Sine and cosecant are positive; cosine, tangent, secant, and cotangent are negative.
• ➗ Quadrant III (x < 0, y < 0): Tangent and cotangent are positive; sine, cosine, secant, and cosecant are negative.
• ✖️ Quadrant IV (x > 0, y < 0): Cosine and secant are positive; sine, tangent, cosecant, and cotangent are negative.
• 🧠 Mnemonic: A common mnemonic to remember which functions are positive in each quadrant is "All Students Take Calculus" (ASTC), representing All (Quadrant I), Sine (Quadrant II), Tangent (Quadrant III), and Cosine (Quadrant IV).
• 💡 r is always positive: Remember that $r = \sqrt{x^2 + y^2}$ is always non-negative. This simplifies determining the sign of the trigonometric functions.
• ✍️ Pay attention to zero and undefined cases: When x = 0 or y = 0, some trigonometric functions will be undefined (division by zero). Be mindful of these cases.

⚙️ Real-World Examples

Example 1: Find the values of the six trigonometric functions for the point (-3, 4).

• 📐 First, find r: $r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
• ✍️ Now, calculate the trigonometric functions:
• sin$\theta = \frac{4}{5}$
• cos$\theta = \frac{-3}{5}$
• tan$\theta = \frac{4}{-3} = -\frac{4}{3}$
• csc$\theta = \frac{5}{4}$
• sec$\theta = \frac{5}{-3} = -\frac{5}{3}$
• cot$\theta = \frac{-3}{4} = -\frac{3}{4}$

Example 2: Find the values of the six trigonometric functions for the point (5, -12).

• 📐 First, find r: $r = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
• ✍️ Now, calculate the trigonometric functions:
• sin$\theta = \frac{-12}{13}$
• cos$\theta = \frac{5}{13}$
• tan$\theta = \frac{-12}{5} = -\frac{12}{5}$
• csc$\theta = \frac{13}{-12} = -\frac{13}{12}$
• sec$\theta = \frac{13}{5}$
• cot$\theta = \frac{5}{-12} = -\frac{5}{12}$

📝 Practice Quiz

Calculate all six trig functions given the following (x, y) coordinates:

1. (3, 4)
2. (-5, 12)
3. (-8, -6)
4. (7, -24)

Answers:

1. sin = 4/5, cos = 3/5, tan = 4/3, csc = 5/4, sec = 5/3, cot = 3/4
2. sin = 12/13, cos = -5/13, tan = -12/5, csc = 13/12, sec = -13/5, cot = -5/12
3. sin = -3/5, cos = -4/5, tan = 3/4, csc = -5/3, sec = -5/4, cot = 4/3
4. sin = -24/25, cos = 7/25, tan = -24/7, csc = -25/24, sec = 25/7, cot = -7/24

🔑 Conclusion

By understanding the relationship between x, y, r, and the quadrants, you can confidently calculate trigonometric functions and avoid common sign errors. Remember the mnemonic and practice regularly to solidify your understanding!