Functions and Trigonometric Functions

Functions

  • A function relates or maps one set of values (input) to another set of values (output).
  • Example: f(x)=x+14f(x) = x + \frac{1}{4} (a straight line).
    • The vertical axis represents f(x)f(x), and the horizontal axis represents xx.
    • When x=0x = 0, f(x)=14f(x) = \frac{1}{4}.

Function Notation

  • f(x)f(x) is read as "f of x," not "f times x."
  • Parentheses are commonly used for the argument of a function, not other brackets.

Trigonometric Functions

  • For simplicity, parentheses are often omitted for simple arguments in trigonometric functions.
    • Instead of sin⁡(θ)\sin(\theta), we often write sin⁡θ\sin \theta.

Sine Function (sin⁡θ\sin \theta)

  • Plotted as a function of angle.
  • sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)

Cosine Function (cos⁡θ\cos \theta)

  • Similar to sine but shifted sideways.
  • cos⁡(0)=1\cos(0) = 1
  • cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta)

Trigonometric Functions and Triangles

  • Defined in terms of right-angled triangles.
    • Base (adjacent side): xx
    • Height (opposite side): yy
    • Hypotenuse: rr

Trigonometric Ratios

  • sin⁡θ=yr\sin \theta = \frac{y}{r}
  • cos⁡θ=xr\cos \theta = \frac{x}{r}
  • tan⁡θ=yx\tan \theta = \frac{y}{x}
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Radians

  • Radians are used for fundamental mathematical calculations.
  • 2π2\pi radians in a circle.
  • 1 radian ≈57.3\approx 57.3 degrees.
  • 360 degrees in a circle.

Other Trigonometric Functions

  • Cosecant: csc⁡θ=1sin⁡θ=ry\csc \theta = \frac{1}{\sin \theta} = \frac{r}{y} (also written as cosec or csc)
  • Secant: sec⁡θ=1cos⁡θ=rx\sec \theta = \frac{1}{\cos \theta} = \frac{r}{x}
  • Cotangent: cot⁡θ=xy=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{x}{y} = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} (also written as cotan or cot)

Inverse Trigonometric Functions

  • sin⁡−1(a)\sin^{-1}(a) (arcsine) is the inverse function of sin⁡θ\sin \theta, giving the angle from the sine value.
  • If a=sin⁡θa = \sin \theta, then arcsin⁡(a)=θ\arcsin(a) = \theta
  • sin⁡−1(a)\sin^{-1}(a) does not mean 1sin⁡a\frac{1}{\sin a} (reciprocal).
  • The superscript -1 indicates the inverse function, not the reciprocal.

Powers of Trigonometric Functions

  • sin⁡2θ\sin^2 \theta means (sin⁡θ)2(\sin \theta)^2 (squaring the function).
  • This notation is mainly used with trigonometric functions.
  • sin⁡2θ\sin^2 \theta is always positive.
  • Similarly, cos⁡2θ\cos^2 \theta means (cos⁡θ)2(\cos \theta)^2.