frontend.js

//Your frontend javascript
var data = {
action: 'sign_in',
username:"<username>",
pass:"<password>"
};

jQuery.post('/wordpress/wp-admin/admin-ajax.php', data, 
	function(response) {
	alert("signed in successfully");
});	

functions.php

//Your functions.php
add_action('wp_ajax_sign_in', 'ajax_sign_in');
add_action('wp_ajax_nopriv_sign_in', 'ajax_sign_in');
function ajax_sign_in() {	
	$creds = array();
	$creds['user_login'] = $_POST['email'];
	$creds['user_password'] = $_POST['pass'];
	$creds['remember'] = true;
	
	$user = wp_signon( $creds, false );
	if ( is_wp_error($user) ) { 
		echo "error"; 
	} else { 
		echo "success";
	}

	die(); // this is required to return a proper result
}