Introduction: Why Study the Unit Circle?
The unit circle is the foundation of trigonometry in Pre-Calculus 40S. It connects geometry, algebra, and real-world applications. Mastery of the unit circle will help you understand trigonometric functions, solve equations, and analyze periodic phenomena in science and engineering.
Visual Guide: The Unit Circle
The unit circle is a circle of radius \(1\) centered at the origin \((0,0)\) in the coordinate plane.
- Equation: \(x^2 + y^2 = 1\)
- Each point \((x, y)\) on the circle corresponds to \((\cos \theta, \sin \theta)\) for some angle \(\theta\).
- Angles are measured from the positive \(x\)-axis, counterclockwise.
Degrees, Radians, and Key Angles
- \(360° = 2\pi \text{ radians}\)
- Common angles: \(0°\), \(30°\), \(45°\), \(60°\), \(90°\), ... up to \(360°\)
$$\text{radians} = \frac{\text{degrees} \times \pi}{180}$$
$$\text{degrees} = \frac{\text{radians} \times 180}{\pi}$$
Key Trigonometric Values on the Unit Circle
| Angle | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| \(0°\) | \(0\) | \(0\) | \(1\) | \(0\) |
| \(30°\) | \(\frac{\pi}{6}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{3}}\) |
| \(45°\) | \(\frac{\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(1\) |
| \(60°\) | \(\frac{\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| \(90°\) | \(\frac{\pi}{2}\) | \(1\) | \(0\) | \(\text{undefined}\) |
Use symmetry and reference angles to find values for other quadrants.
Coterminal Angles
Coterminal angles are angles that share the same initial and terminal sides, but may have different measures. You can find coterminal angles by adding or subtracting full rotations.
$$\text{coterminal angle} = \theta + 360^\circ \times k$$ where k is any integer
$$\text{coterminal angle} = \theta + 2\pi k$$ where k is any integer
- To find a positive coterminal angle, add \(360°\) (or \(2\pi\)) as many times as needed based on the question's given range.
- To find a negative coterminal angle, subtract \(360°\) (or \(2\pi\)) as many times as needed based on the question's given range.
Trigonometric Equations
Trigonometric equations involve unknown angles and trigonometric functions. To solve them, isolate the trig function and use inverse operations, considering all possible solutions in the given interval.
Solve for \(\theta\) in \([0^\circ, 360^\circ]\):
$$5\cos\theta + 7 = 4$$
- Isolate \(\cos\theta\):
$$5\cos\theta = 4 - 7 = -3$$
$$\cos\theta = \frac{-3}{5}$$ - Find all \(\theta\) such that \(\cos\theta = -0.6\) in \([0^\circ, 360^\circ]\):
$$\theta = \cos^{-1}(-0.6)$$
Calculator gives: \(\theta_1 \approx 126.87^\circ\) - Since cosine is negative in Quadrants II and III, the second solution is:
$$\theta_2 = 360^\circ - 126.87^\circ = 233.13^\circ$$
\(\boxed{\theta \approx 126.87^\circ,\ 233.13^\circ}\)
Always check the interval and consider all possible solutions for the trigonometric function.
Key Formulas & Identities
- Pythagorean Identity: \(sin^2θ + cos^2θ = 1\)
- \(sin(−θ) = −sin θ, cos(−θ) = cos θ\)
- \(sin(π − θ) = sin θ, cos(π − θ) = −cos θ\)
- \(tan θ = \frac{sin θ}{cos θ}\)
Finding the Length of an Arc on a Circle
The length of an arc (a portion of the circumference) depends on the angle at the center and the radius of the circle.
- If the angle is in degrees:
Arc Length = (Angle / 360) × 2πr
where r = radius, angle in degrees - If the angle is in radians:
Arc Length = r × θ
where r = radius, θ = angle in radians
These formulas are essential for solving problems involving sectors and arcs in trigonometry and geometry.
Common Mistakes & How to Avoid Them
- Confusing degrees and radians—always check units!
- Forgetting the signs of trig functions in different quadrants.
- Not using reference angles for non-standard positions.
- Mixing up sine and cosine coordinates.