Honors Algebra II Unit 11 Review: Trigonometric Functions and Identities

Key Concepts and Definitions

• Trigonometry studies relationships between side lengths and angles in triangles
• Sine ($\sin$), cosine ($\cos$), and tangent ($\tan$) are the primary trigonometric functions
  • $\sin$ is the ratio of the opposite side to the hypotenuse
  • $\cos$ is the ratio of the adjacent side to the hypotenuse
  • $\tan$ is the ratio of the opposite side to the adjacent side
• Cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$) are reciprocal functions of $\sin$, $\cos$, and $\tan$, respectively
• Trigonometric identities are equations that hold true for all values of the variable
• Radian measure is an alternative to degrees for measuring angles
  • One radian is the angle subtended by an arc length equal to the radius of a circle
• The unit circle has a radius of 1 and is centered at the origin of a coordinate plane

Angles and Radian Measure

• Angles can be measured in degrees or radians
• To convert from degrees to radians, multiply by $\frac{\pi}{180}$
• To convert from radians to degrees, multiply by $\frac{180}{\pi}$
• Angles in standard position have their vertex at the origin and initial side along the positive x-axis
• Coterminal angles share the same terminal side and differ by multiples of $2\pi$ radians or 360 degrees
• Reference angles are acute angles formed by the terminal side of an angle and the x-axis
• Quadrantal angles are multiples of $\frac{\pi}{2}$ radians or 90 degrees ($0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$)
• Angles can be positive (counterclockwise) or negative (clockwise)

The Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane
• Angles in the unit circle are measured in radians, with 0 at the point (1, 0) and increasing counterclockwise
• The coordinates of a point on the unit circle are ($\cos \theta$, $\sin \theta$), where $\theta$ is the angle from the positive x-axis
• The unit circle is divided into four quadrants, each spanning $\frac{\pi}{2}$ radians or 90 degrees
• Trigonometric functions have specific values at key angles on the unit circle ($0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$)
  • These values can be derived using special right triangles (30-60-90 and 45-45-90)
• Memorizing the unit circle helps simplify trigonometric calculations and graphing

Graphing Trigonometric Functions

• The basic trigonometric functions ($\sin$, $\cos$, $\tan$) have distinct graph shapes
  • $\sin$ and $\cos$ graphs are wavy and repeat every $2\pi$ radians
  • $\tan$ graph has vertical asymptotes at odd multiples of $\frac{\pi}{2}$ and repeats every $\pi$ radians
• Graphs can be transformed by changing the amplitude, period, phase shift, and vertical shift
  • Amplitude is the vertical stretch or compression of the graph
  • Period is the horizontal width of one complete cycle
  • Phase shift is the horizontal translation of the graph
  • Vertical shift is the vertical translation of the graph
• The general forms of trigonometric functions are:
  • $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$
  • $A$ is amplitude, $B$ is frequency ($\frac{2\pi}{\text{period}}$), $C$ is phase shift, and $D$ is vertical shift
• To graph, plot key points using the transformed period, amplitude, and shifts, then connect the points smoothly

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all input values
• Pythagorean identities relate $\sin^2 \theta$, $\cos^2 \theta$, and 1
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $1 + \cot^2 \theta = \csc^2 \theta$
• Angle addition and subtraction formulas express $\sin$, $\cos$, or $\tan$ of a sum or difference of angles
  • Example: $\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
• Double-angle formulas express $\sin$, $\cos$, or $\tan$ of twice an angle in terms of the original angle
  • Example: $\sin(2\theta) = 2\sin \theta \cos \theta$
• Half-angle formulas express $\sin$, $\cos$, or $\tan$ of half an angle in terms of the original angle
  • Example: $\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$
• Identities can be used to simplify expressions, solve equations, and prove other identities

Solving Trigonometric Equations

• To solve trigonometric equations, isolate the trigonometric function on one side of the equation
• If the equation involves a single trigonometric function, use the inverse function to solve
  • Example: If $\sin \theta = \frac{1}{2}$, then $\theta = \arcsin(\frac{1}{2})$
• If the equation involves multiple trigonometric functions, use identities to rewrite the equation in terms of a single function
• Consider the domain and range of the functions involved to determine the number of solutions
• Remember to include solutions from all periods (add multiples of $2\pi$ for $\sin$ and $\cos$, or $\pi$ for $\tan$)
• Verify solutions by substituting them back into the original equation
• Some equations may require factoring or using the quadratic formula to solve

Real-World Applications

• Trigonometry is used in various fields, such as physics, engineering, and navigation
• Right triangle trigonometry can be used to find unknown lengths or angles in real-world objects (buildings, bridges)
• Periodic phenomena, such as sound waves, tides, and pendulum motion, can be modeled using trigonometric functions
• In physics, trigonometric functions are used to resolve vectors into components (forces, velocities)
• Trigonometry is essential for understanding and manipulating complex numbers, which have applications in electrical engineering and quantum mechanics
• Trigonometric functions are used in geodesy and satellite navigation to calculate distances and positions on the Earth's surface
• In computer graphics and game development, trigonometric functions are used for rotation, scaling, and animation

Common Pitfalls and Tips

• Remember the order of operations when simplifying trigonometric expressions (PEMDAS)
• Be careful with the signs of angles in different quadrants (ASTC: All Students Take Calculus)
• When using inverse trigonometric functions, consider the domain and range to determine the appropriate quadrant for the solution
• Double-check that your calculator is in the correct mode (degrees or radians) before performing calculations
• Sketch graphs to visualize problems and check the reasonableness of your solutions
• Memorize the unit circle and common angle values to save time during tests and quizzes
• Practice regularly to maintain your skills and understanding of trigonometric concepts
• When solving equations, make sure to solve for the variable, not just the trigonometric function
• Remember that $\sin$ and $\cos$ are even functions, while $\tan$ is an odd function