diff --git a/problem_set/Q2_Chinese_Zodiac.py b/problem_set/Q2_Chinese_Zodiac.py
index 010f431..46ac82e 100644
--- a/problem_set/Q2_Chinese_Zodiac.py
+++ b/problem_set/Q2_Chinese_Zodiac.py
@@ -10,4 +10,29 @@
 
 # write a function that takes the year as the input and produces the Zodiac for that year
 # output should be in the format "2000 is the year of the Fire Rat"
+chinese_dict= {
+    'animals': ["Rat","Ox","Tiger","Rabbit","Dragon","Snake","Horse","Goat","Monkey","Rooster","Dog","Pig"],
+    'elements': ["Fire", "Earth", "Metal", "Water", "Wood"]
+}
+year=2000
+def chinese_new_year(x):
+    """
+    With an inputted year, the function will determine the CHinese Zodiac associated with that year
+    in a initialized cycle.
+    """
+    x = int(x)
+    year = (x-2024) % 60
+    result=[]
+    for element in chinese_dict["elements"]:
+        for animal in chinese_dict["animals"]:
+            order = f"{element} {animal}"
+            result.append(order)
+    Zodiac = result[year]
+    if x<2024:
+        Messege = f"{x} was the year of the {Zodiac}"
+    else:
+        Messege = f"{x} is the year of the {Zodiac}"
+    return Messege
 
+Zodiac = chinese_new_year(input("Input a year to determine it's associated Zodiac: "))
+print(Zodiac)
\ No newline at end of file
diff --git a/problem_set/Q3_Newton_Raphson_method.py b/problem_set/Q3_Newton_Raphson_method.py
index c6bcbc2..5354adb 100644
--- a/problem_set/Q3_Newton_Raphson_method.py
+++ b/problem_set/Q3_Newton_Raphson_method.py
@@ -2,24 +2,98 @@
 
 # Use this root finding method to evaluate the square root of 2 to 6 decimal places
 
-# The square root of 2 is 1.414213
+# The square root of 2 is 1.414213 or 1.4142135623730951
 
 true_value = 1.414213
-
+import math
 # compare your answer with the true value
 
 # Define the function whose root we are trying to find
 
-
 ### write comments and code here ###
+## Tool 1
+def function_x(x):
+    """
+    Define the Parent Function
+    """
+    y= x**2 - 2
+    return y
+# print(newton_rahpson(math.pi/2))
 
 
-# Define the derivative of the function
+# def RootstoQuadratic(x1,x2):
+#     """
+#     Using the knowledge of what a factored Quadratic looks like: x1,x2 = x² -x(x1+x2) + x1*x2.
+#     We can descrivbe a quadratic equation for any two inputted roots (in the Real plane).
+#     a = 1 because the value of c may compensate for the distances between the roots. 
+#     (I hope to fix the issue and optimize for using a so that the value of c doesn't erupt.)
+#     """
+#     a= 1
+#     b= -(x1+x2)
+#     c= (x1*x2)
+#     equation = f"x² "
+#     if b == 0:
+#         equation = f"x² "
+#     else:
+#         if b < 0:
+#             equation += f"- {-b}x "
+#         else:
+#             equation += f"+ {b}x "
 
+#     if c < 0:
+#         equation += f"- {-c}"
+#     else:
+#         equation += f"+ {c}"
+#     if c == 0:
+#         equation = f"x²"
+#     print(f"The equation for these roots is: {equation}")
+#     return a,b,c
+# a,b,c = RootstoQuadratic((3/2),6)
 
-### write comments and code here ###
+# # Define the derivative of the function
+
+# ### write comments and code here ###
+# ## Tool 2
+
+def derivative_of_f(x):
+    """
+    Define the derivative of the parent function
+    """
+    y_prime= 2*(x)
+    return y_prime
+
+
+# """
+# We have an advantage because we always know we are dealing with a quadratic.
+# It will be: 2x -(x1 + x2)
+# """
+# def QuadraticDerivative(b):
+#     if b != 0:
+#         if b < 0:
+#             print(f"The derivative is: 2x - {-b}")
+#         else:
+#             print(f"The derivative is: 2x + {b}")
+#     else:
+#         print("The derivative is: 2x")
+#     return b
+# QuadraticDerivative(b)
 
 
-# Run the algorithm for the necessary amount of iterations to converge to 6 d.p. accuracy, make sure to start with a good initial guess
+# # Run the algorithm for the necessary amount of iterations to converge to 6 d.p. accuracy, make sure to start with a good initial guess
+# ## Newton_Raphson_method
+## xn+1 = xn - f(x)/f'(x)
 
+def newton_rahpson(guess: int):
+    """
+    This function uses a parent and derivative to approximate a root given an initial guess.
+    It is specified to a certain error.
+    """
+    y = 1
+    while abs(y) > .000001:
+        next_guess = guess - (function_x(guess)/derivative_of_f(guess))
+        guess=next_guess
+        y = function_x(next_guess)
+    return next_guess
 
+print(newton_rahpson(5))
+print(f"True Value: {true_value}")
\ No newline at end of file
diff --git a/problem_set/Q4_prime_number.py b/problem_set/Q4_prime_number.py
index 00057b7..d39109a 100644
--- a/problem_set/Q4_prime_number.py
+++ b/problem_set/Q4_prime_number.py
@@ -9,3 +9,33 @@
 # Do not use a naive approach of checking every single number below
 # At the least, do not check even numbers
 
+def prime_number_solver(number):
+    """
+    This function takes an input number and tests if it's prime or composite.
+    Outputs True or False respectfully.
+    """
+    divisor = 3
+    sqrt_number = number**.5
+    compute = 0
+    #check if even: num % 2 == 0
+    if number == 1:
+        print("1 is neither prime nor composite!")
+        return None
+    elif number == 2:
+        print("2 is a prime number!")
+        return False
+    else:
+        if number % 2 == 0:
+            print(f"{number} is composite! It is even!")
+            return False
+    while divisor <= sqrt_number:
+        compute = number % divisor
+        if compute == 0:
+            print(f"{number} is composite! It failed with {divisor}.")
+            return False
+        divisor += 2
+    else:
+        print(f"{number} is prime!")
+        return True
+x = prime_number_solver(number)
+print(x)
\ No newline at end of file
diff --git a/problem_set/Q5_taylor_series.py b/problem_set/Q5_taylor_series.py
index 8e2b6ad..03e5163 100644
--- a/problem_set/Q5_taylor_series.py
+++ b/problem_set/Q5_taylor_series.py
@@ -5,5 +5,35 @@
 # Use numpy and matplotlib to plot your results and display them in the colour blue
 
 # plot the real value of f(x) = sin(x) on the same plot in the colour red
-# 
-#  
+
+import matplotlib.pyplot as plt
+import numpy as np
+import math as math
+
+def Taylor_Series(x):
+    """
+    This funciton takes an input (x) with a defined taylor series function (in this case sin(x)).
+    It outputs a value, y, after summing a specified range of the series (n).
+    """
+    sum=0
+    for n in range(0,11):
+        y = ((-1)**n * x**(2*n+1))/(math.factorial(2*n+1))
+        sum += y
+    return sum
+
+x_range = np.arange(-2*math.pi,2*math.pi,.1) ## Makes the x range for the plot
+output1 = [] ## Initial list for our taylor series function
+for i in x_range:
+    y = Taylor_Series(i) ##Iterates through taylor series with the defined domain
+    output1.append(y)
+y_range1 = np.array(output1) ##This is the y range for the taylor approximation in an array format to plot
+y_range2 = np.sin(x_range) ##This is the second graph of the true sin(x) function to compare with Taylor
+
+fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 5)) ##This creates space for 2 plots##
+
+ax1.plot(y_range1, color = 'blue') ##First plot of Taylor Approximation sin(x)
+
+ax2.plot(y_range2, color = 'red')##Second plot of true sin(x)
+
+plt.tight_layout()##formatting function
+plt.show()
\ No newline at end of file