How can I get sin, cos, and tan to use degrees instead of radians?

You can use a function like this to do the conversion:

function toDegrees (angle) {
  return angle * (180 / Math.PI);
}

Note that functions like sin, cos, and so on do not return angles, they take angles as input. It seems to me that it would be more useful to you to have a function that converts a degree input to radians, like this:

function toRadians (angle) {
  return angle * (Math.PI / 180);
}

which you could use to do something like tan(toRadians(45)).

I created my own little lazy Math-Object for degree (MathD), hope it helps:

//helper
/**
 * converts degree to radians
 * @param degree
 * @returns {number}
 */
var toRadians = function (degree) {
    return degree * (Math.PI / 180);
};

/**
 * Converts radian to degree
 * @param radians
 * @returns {number}
 */
var toDegree = function (radians) {
    return radians * (180 / Math.PI);
}

/**
 * Rounds a number mathematical correct to the number of decimals
 * @param number
 * @param decimals (optional, default: 5)
 * @returns {number}
 */
var roundNumber = function(number, decimals) {
    decimals = decimals || 5;
    return Math.round(number * Math.pow(10, decimals)) / Math.pow(10, decimals);
}
//the object
var MathD = {
    sin: function(number){
        return roundNumber(Math.sin(toRadians(number)));
    },
    cos: function(number){
        return roundNumber(Math.cos(toRadians(number)));
    },
    tan: function(number){
        return roundNumber(Math.tan(toRadians(number)));
    },
    asin: function(number){
        return roundNumber(toDegree(Math.asin(number)));
    },
    acos: function(number){
       return roundNumber(toDegree(Math.acos(number)));
   },
   atan: function(number){
       return roundNumber(toDegree(Math.atan(number)));
   }
};

Multiply the input by Math.PI/180 to convert from degrees to radians before calling the system trig functions.

You could also define your own functions:

function sinDegrees(angleDegrees) {
    return Math.sin(angleDegrees*Math.PI/180);
};

and so on.