1.3 Trigonometric Functions

Angle measurements and trigonometric functions are essential tools in calculus. They help us understand circular motion and periodic behavior. We'll explore how degrees and radians measure angles, and how sine, cosine, and tangent relate to triangles and circles.

Trigonometric identities and graphs build on these foundations. We'll learn key relationships between trig functions and how to visualize them. Understanding how parameters affect these graphs is crucial for modeling real-world phenomena in calculus.

Angle Measurements and Trigonometric Functions

Degree vs radian measurements

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• Degrees divide a full circle into 360 equal parts ($360^\circ$)
  • $1^\circ$ represents $\frac{1}{360}$ of a full circle rotation
  • Commonly used in everyday life (compass directions, weather reports)
• Radians measure angles based on the radius of a circle
  • 1 radian is the angle subtended by an arc length equal to the radius
  • $2\pi$ radians correspond to a full circle ($360^\circ$)
  • Mathematically convenient for calculus and physics
• Convert degrees to radians: $\text{radians} = \text{degrees} \cdot \frac{\pi}{180}$
• Convert radians to degrees: $\text{degrees} = \text{radians} \cdot \frac{180}{\pi}$

Trigonometric functions in triangles and circles

• Right triangle definitions (SOH-CAH-TOA mnemonic):
  • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ (ratio of opposite side to hypotenuse)
  • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ (ratio of adjacent side to hypotenuse)
  • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ (ratio of opposite side to adjacent side)
• Unit circle definitions ($r=1$ centered at origin):
  • $\sin(\theta) = y\text{-coordinate}$ of point on circle at angle $\theta$
  • $\cos(\theta) = x\text{-coordinate}$ of point on circle at angle $\theta$
  • $\tan(\theta) = \frac{y\text{-coordinate}}{x\text{-coordinate}}$ of point on circle at angle $\theta$
  • Angle $\theta$ measured counterclockwise from positive x-axis

Properties of Trigonometric Functions

• Domain: The set of all possible input values (angles) for which the function is defined
• Range: The set of all possible output values of the function
• Periodicity: The repetition of function values over regular intervals of the input (angle)
• Each trigonometric function is a mathematical relationship between an angle and a ratio of sides of a right triangle

Trigonometric Identities and Graphs

Applications of trigonometric identities

• Reciprocal identities express trigonometric functions in terms of their reciprocals:
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$ (cosecant is reciprocal of sine)
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$ (secant is reciprocal of cosine)
  • $\cot(\theta) = \frac{1}{\tan(\theta)}$ (cotangent is reciprocal of tangent)
• Pythagorean identities relate trigonometric functions based on the Pythagorean theorem:
  • $\sin^2(\theta) + \cos^2(\theta) = 1$ (sine squared plus cosine squared equals 1)
  • $1 + \tan^2(\theta) = \sec^2(\theta)$ (1 plus tangent squared equals secant squared)
  • $1 + \cot^2(\theta) = \csc^2(\theta)$ (1 plus cotangent squared equals cosecant squared)

Graphs of basic trigonometric functions

• Sine function $y = \sin(\theta)$:
  • Periodic repeating every $2\pi$ radians ($360^\circ$)
  • Oscillates between -1 and 1 (range [-1, 1])
  • Odd symmetry $\sin(-\theta) = -\sin(\theta)$ (graph is symmetric about origin)
• Cosine function $y = \cos(\theta)$:
  • Periodic repeating every $2\pi$ radians ($360^\circ$)
  • Oscillates between -1 and 1 (range [-1, 1])
  • Even symmetry $\cos(-\theta) = \cos(\theta)$ (graph is symmetric about y-axis)
• Tangent function $y = \tan(\theta)$:
  • Periodic repeating every $\pi$ radians ($180^\circ$)
  • Unbounded range $(-\infty, \infty)$ (can take on any real value)
  • Odd symmetry $\tan(-\theta) = -\tan(\theta)$ (graph is symmetric about origin)
  • Vertical asymptotes at $\theta = \frac{\pi}{2} + n\pi$ for integer $n$ ($90^\circ$, $270^\circ$, etc.)

Effects of parameters on trigonometric graphs

• Amplitude $A$ vertically stretches ($A>1$) or compresses ($0<A<1$) the graph:
  • $y = A\sin(\theta)$ or $y = A\cos(\theta)$
• Period $\frac{2\pi}{B}$ horizontally compresses ($B>1$) or stretches ($0<B<1$) the graph:
  • $y = \sin(B\theta)$ or $y = \cos(B\theta)$
• Phase shift $\frac{C}{B}$ horizontally shifts the graph left ($C>0$) or right ($C<0$):
  • $y = \sin(B\theta + C)$ or $y = \cos(B\theta + C)$
• Vertical shift $D$ translates the graph up ($D>0$) or down ($D<0$):
  • $y = \sin(\theta) + D$ or $y = \cos(\theta) + D$