🍬Honors Algebra II Unit 12 – Trigonometric Equations & Applications

Key Concepts and Definitions

• Trigonometric functions relate angles to ratios of side lengths in a right triangle
• Sine ($\sin$), cosine ($\cos$), and tangent ($\tan$) are the primary trigonometric functions
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Reciprocal functions include cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$)
• Radian measure expresses angles as the ratio of arc length to radius on the unit circle
  • $2\pi$ radians equals 360 degrees
• Periodic functions repeat their values at regular intervals called periods
• Amplitude measures the height of a function's maximum oscillation from its midline
• Phase shift describes the horizontal displacement of a function from its standard position

Trigonometric Functions Review

• Trigonometric functions have specific values at key angles (0°, 30°, 45°, 60°, 90°)
• Sine and cosine functions oscillate between -1 and 1
  • $\sin 0° = 0$, $\sin 90° = 1$
  • $\cos 0° = 1$, $\cos 90° = 0$
• Tangent function has a period of $\pi$ and asymptotes at odd multiples of $\frac{\pi}{2}$
• Reciprocal functions are the multiplicative inverses of their counterparts
  • $\csc \theta = \frac{1}{\sin \theta}$
  • $\sec \theta = \frac{1}{\cos \theta}$
  • $\cot \theta = \frac{1}{\tan \theta}$
• Pythagorean identities relate trigonometric functions
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $1 + \cot^2 \theta = \csc^2 \theta$

Solving Basic Trig Equations

• Isolate the trigonometric function on one side of the equation
• Determine the angle that satisfies the equation using inverse trigonometric functions
  • $\sin^{-1}$ (arcsine), $\cos^{-1}$ (arccosine), $\tan^{-1}$ (arctangent)
• Consider the domain and range of the functions to find all possible solutions
• Use the unit circle to visualize solutions and determine angle measures
• Remember to express answers in the desired unit (degrees or radians)
• Check solutions by substituting them back into the original equation

Advanced Trig Equation Techniques

• Factoring can simplify trigonometric equations and reveal solutions
• Pythagorean identities can be used to rewrite equations in terms of a single function
• Double-angle formulas express $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ in terms of $\theta$
  • $\sin 2\theta = 2\sin \theta \cos \theta$
  • $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
  • $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$
• Half-angle formulas express $\sin \frac{\theta}{2}$, $\cos \frac{\theta}{2}$, and $\tan \frac{\theta}{2}$ in terms of $\theta$
• Sum and difference formulas expand $\sin(\alpha \pm \beta)$, $\cos(\alpha \pm \beta)$, and $\tan(\alpha \pm \beta)$
• Solve equations involving multiple angles by applying these formulas strategically

Graphing Trig Functions

• Sine and cosine functions have a period of $2\pi$, while tangent has a period of $\pi$
• Amplitude affects the vertical stretch or compression of the graph
  • $f(x) = A\sin x$ has an amplitude of $|A|$
• Phase shift moves the graph horizontally
  • $f(x) = \sin(x - \phi)$ shifts the graph to the right by $\phi$ units
• Vertical shift moves the graph up or down
  • $f(x) = \sin x + k$ shifts the graph up by $k$ units
• Period changes affect the horizontal stretch or compression of the graph
  • $f(x) = \sin(Bx)$ has a period of $\frac{2\pi}{|B|}$
• Combine transformations to create more complex trigonometric functions

Real-World Applications

• Modeling periodic phenomena (tides, sound waves, pendulums, etc.)
• Calculating heights and distances using angle measurements (triangulation)
• Analyzing harmonic motion in physics and engineering
  • Mass-spring systems, simple pendulums
• Describing electrical signals and alternating currents
• Predicting cyclic patterns in economics, biology, and social sciences
  • Population dynamics, business cycles, circadian rhythms
• Solving problems involving navigation and surveying
  • Bearing, elevation, and depression angles

Common Pitfalls and Tips

• Remember to work in the correct unit (degrees or radians)
• Be cautious when using inverse trigonometric functions
  • Restrict the domain to get the desired solution
• Consider the periodicity of functions when solving equations
  • Solutions may exist in multiple periods
• Sketch graphs to visualize transformations and identify key features
• Double-check signs and quadrants when evaluating trigonometric functions
• Simplify expressions and equations before attempting to solve them
• Verify that solutions satisfy the original equation by substituting them back in

Practice Problems and Examples

1. Solve for $\theta$: $2\sin \theta = \sqrt{3}$
   • $\sin \theta = \frac{\sqrt{3}}{2}$
   • $\theta = \sin^{-1}(\frac{\sqrt{3}}{2})$ or $\theta = \frac{\pi}{3}$
2. Find all solutions to $\cos 2x = \frac{1}{2}$ on the interval $[0, 2\pi]$
   • Using the double-angle formula: $\cos^2 x - \sin^2 x = \frac{1}{2}$
   • Solve for $x$ using the Pythagorean identity and inverse cosine
   • Solutions: $x = \frac{\pi}{6}, \frac{5\pi}{6}$
3. Graph the function $f(x) = 3\sin(2x - \frac{\pi}{4}) + 1$
   • Amplitude: 3
   • Period: $\frac{2\pi}{2} = \pi$
   • Phase shift: $\frac{\pi}{8}$ to the right
   • Vertical shift: 1 unit up
4. A Ferris wheel with a radius of 50 feet makes one rotation every 2 minutes. Write an equation for the height $h$ of a rider as a function of time $t$ (in seconds).
   • $h(t) = 50\sin(\frac{\pi}{60}t) + 50$
5. Prove the identity: $\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x}$
   • Left side: $\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \tan x + \cot x$
   • Right side: $\frac{1}{\sin x \cos x} = \frac{1}{\frac{1}{2}\sin 2x} = \frac{2}{\sin 2x}$
   • Using the double-angle formula: $\frac{2}{\sin 2x} = \frac{2}{2\sin x \cos x} = \frac{1}{\sin x \cos x}$