🔺Trigonometry Unit 1 – Trigonometric Functions

Key Concepts and Definitions

• Trigonometry studies relationships between side lengths and angles of triangles
• Sine, cosine, and tangent are the primary trigonometric functions
  • Sine ($\sin$) is the ratio of the opposite side to the hypotenuse
  • Cosine ($\cos$) is the ratio of the adjacent side to the hypotenuse
  • Tangent ($\tan$) is the ratio of the opposite side to the adjacent side
• Reciprocal functions include cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$)
• Radian measure expresses angle size as the ratio of arc length to radius
• Periodic functions repeat their values at regular intervals (period)
• Amplitude measures the height of a function's graph from its midline

Angles and Radian Measure

• Angles can be measured in degrees or radians
  • 360 degrees or $2\pi$ radians make up a full rotation
• Radian measure relates angle size to the length of an arc on the unit circle
  • One radian is the angle subtended by an arc length equal to the radius
• To convert between degrees and radians, use the formulas:
  • $\text{radians} = \text{degrees} \cdot \frac{\pi}{180}$
  • $\text{degrees} = \text{radians} \cdot \frac{180}{\pi}$
• 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 (differ by multiples of $2\pi$ radians or 360 degrees)
• Reference angles are acute angles formed by the terminal side and x-axis

The Unit Circle

• The unit circle has a radius of 1 and is centered at the origin
• Angle measures are placed around the circumference in radians or degrees
• Coordinates of points on the unit circle are $(\cos\theta, \sin\theta)$
  • $\cos\theta$ is the x-coordinate, and $\sin\theta$ is the y-coordinate
• Special angles ($0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$) have exact trigonometric values
• Trigonometric functions have symmetry properties based on the unit circle
  • $\sin(-\theta) = -\sin(\theta)$ (odd function)
  • $\cos(-\theta) = \cos(\theta)$ (even function)

Trigonometric Functions and Their Graphs

• Sine, cosine, and tangent functions are periodic and have distinct graphs
• Sine function: $f(x) = \sin(x)$
  • Period: $2\pi$, Amplitude: 1, Range: [-1, 1]
• Cosine function: $f(x) = \cos(x)$
  • Period: $2\pi$, Amplitude: 1, Range: [-1, 1]
• Tangent function: $f(x) = \tan(x)$
  • Period: $\pi$, Range: $(-\infty, \infty)$, undefined at $x = \frac{\pi}{2} + \pi n$
• Graphs can be transformed by changing amplitude, period, phase shift, or vertical shift
  • $A \sin(B(x - C)) + D$ or $A \cos(B(x - C)) + D$
• Reciprocal functions (cosecant, secant, cotangent) have unique graphs and properties

Trigonometric Identities

• Trigonometric identities are equations that hold true for all values of the variable
• Pythagorean identities: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, $1 + \cot^2(\theta) = \csc^2(\theta)$
• Angle sum and difference identities for sine, cosine, and tangent
  • Example: $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
• Double-angle and half-angle identities
  • Example: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
• Identities can simplify expressions and solve equations

Solving Trigonometric Equations

• Trigonometric equations involve trigonometric functions and can be solved for the variable
• Isolate the trigonometric function and use inverse functions (e.g., $\sin^{-1}, \cos^{-1}, \tan^{-1}$)
  • Consider the domain and range of inverse functions
• Solve equations over a specific interval (e.g., $[0, 2\pi]$) to find all solutions
• Factoring, identities, or graphing can help solve more complex equations
• Be aware of extraneous solutions introduced by transformations

Applications in Real-World Problems

• Trigonometry has applications in various fields (physics, engineering, navigation)
• Right triangle problems: use trigonometric ratios (SOH-CAH-TOA) to find unknown sides or angles
• Angle of elevation and depression problems
  • Elevation: angle above horizontal, Depression: angle below horizontal
• Harmonic motion problems (springs, pendulums) involve sinusoidal functions
• Trigonometry aids in analyzing periodic phenomena (tides, sound waves, electrical signals)

Common Pitfalls and Tips

• Remember the order of operations (PEMDAS) when simplifying expressions
• Be cautious when using inverse trigonometric functions
  • Check the domain and range, and consider quadrants
• Sketch diagrams to visualize problems and identify given information
• Double-check signs (positive or negative) when using trigonometric ratios
• Memorize common angle values and identities to save time during tests
• Practice problems with various difficulty levels to reinforce concepts
• Utilize resources (textbooks, online tutorials, study groups) for extra help