Inverse Trigonometric Functions With Sass

trig.scss

// ----
// Sass (v3.3.7)
// Compass (v1.0.0.alpha.18)
// ----

/**
 * Inverse Trigonometric Functions With Sass
 *
 * Functions that compute the values for angles whose 
 * sine, cosine, or tangent is known.
 *
 * http://thesassway.com/advanced/inverse-trigonometric-functions-with-sass
 */
$pi: pi();
$default-threshold: $pi/180/10;

//
// ANGLE CONVERSION HELPER
//
@function convert-angle($value, $unit-name) {
  $factors: (
    rad: 1rad,
    deg: 180deg / $pi,
    turn: .5turn / $pi,
    grad: 200grad / $pi
  );
  
  @if not unitless($value) {
    @warn '\n`#{$value}` should be unitless\n';
    @return false;
  }
  
  @if not map-has-key($factors, $unit-name) {
    @warn '\nUnit `#{$unit-name}` is not a valid unit. \nPlease make sure it is either `deg`, `rad`, `grad` or `turn`.\n';
    @return false;
  }
  
  @return $value * map-get($factors, $unit-name);
}

//
// CALCULATE ARCSINE
// given an angle's sine (z), 
// this fn will tell you the angle
//
@function asin($z, $unit-name: deg, $threshold: $default-threshold) {
  $sum: 0;
  $is-complement: false;
  
  // make sure it works for negative $z values as well
  $sign: $z / abs($z); // -1 or 1
  $z: abs($z); // always positive
  
  @if $z > sin($pi/4) {
    $is-complement: true;
    $z: sqrt(1 - pow($z, 2));
  }
  
  // add up terms to the sum and set the condition that we 
  // stop once we got to a term whose value is smaller 
  // than the threshold value which we pass to the function
  $term: $z;
  $i: 0; // current term's index
  $k: 1; // previous term used in @while (below)
  
  @if abs($z) <= 1 {
    @while $term > $threshold {
      $sum: $sum + $term;
      
      $i: $i + 1; // iterate
      $k: $k * (2 * $i - 1) / (2 * $i); // store previous term
      $j: 2 * $i + 1;
      
      $term: $k * pow($z, $j) / $j;
    }
  } @else {
    @warn '\nabs(#{$z}) must be <= 1 in order for $term to get under the $threshold. \nInfinite Loop would result.\n';
    
    @return false;
  }
  
  // multiply result by $sign again to ensure a 
  // negative number remains that way after computations
  @return convert-angle($sign * (if($is-complement, $pi/2 - $sum, $sum)));
}

//
// CALCULATE ARCCOSINE
// In the case of an angle α in the [0,π] interval, cos(α)=sin(π/2−α).
// If we know cos(α)=z, then arcsin(z)=π/2−α, 
// which gives us that α=π/2−arcsin(z)
//
@function acos($z, $unit-name: deg, $threshold: $default-threshold) {
  @return convert-angle($pi/2 - asin($z, $threshold));
}

//
// CALCULATE ARCTANGENT
// tan(α)=sin(α)/cos(α). We also know that sin2(α)+cos2(α)=1, 
// so we have that tan2(α)=sin2(α)/(1−sin2(α)). 
// From this equality, we extract the sine depending on the tangent.
//
@function atan($z, $unit-name: deg, $threshold: $default-threshold) {
  @return asin($z / sqrt(1 + pow($z, 2)), $threshold);
}