MATH 135 Waterloo

MATH 135 (Algebra for Honours Mathematics) is one of the most difficult first year courses in University of Waterloo. This course covers the following topics:

  • Logic, proofs, mathematical induction
  • Divisibility, primes, GCD
  • Extended Euclidean algorithm, linear Diophantine equations
  • Linear congruences, Fermat’s little theorem, Chinese remainder theorem
  • Public key cryptography, RSA, including fast exponentiation
  • Complex numbers, the complex plane, polar representation
  • De Moivre’s theorem, Fundamental theorem of algebra
  • Polynomials, factorization, roots, error-correcting codes
  • Equations over finite fields, partial fractions

To be successful in MATH 135 you really need to know the theorems, definitions and methods very well. The proofs are most important part for Math 135.

If you need help Math 135, contact us.

Half angle formula


Trig identities are crucial part of trig functions and one of the most important identities in trigonometric functions is the half angle formula. In this post, we talked about different trig identities and formulas. First we recommend to visit the previous posts to review identities, including the double angle formula. Here we talk about half angle formulas. We introduce them, then we see how to use them with some examples and applications.

What is the half angle formula?

There are 3 main half angle formulas for sin, cos and tan:

  • $\sin^{2} u=\frac{1-\cos (2 u)}{2}$
  • $\cos^{2} u=\frac{1+\cos (2 u)}{2}$
  • $\tan^{2} u=\frac{1-\cos (2 u)}{1+\cos (2 u)}$

Next we see some good and useful examples of half angle formula. First, try to solve them by yourself and then check the solutions carefully.

Examples of half angle formula:

Example 1: Find $\sin (\dfrac{\pi}{12})$.

Solution: To solve this problem, we use the half angle formula for the function $\sin$.
$$\sin^{2}(\dfrac{\pi}{12})=\frac{1-\cos (2 \dfrac{\pi}{12})}{2}=\frac{1-\cos (\dfrac{\pi}{6})}{2}$$
We know that $\cos (\dfrac{\pi}{6})=\dfrac{\sqrt{3}}{2}$. So $$\sin^{2}(\dfrac{\pi}{12})=\dfrac{1-\dfrac{\sqrt{3}}{2}}{2}=\dfrac{2-\sqrt{3}}{4}$$
Since $\pi/12$ is in the first quadrant, then $\sin (\dfrac{\pi}{12})$ is positive, so
$$\sin^{2}(\dfrac{\pi}{12})=\sqrt{ \dfrac{2-\sqrt{3}}{4} }$$

Example 2: Prove the identity $2\cos^2 x \sec(2x)=\sec(2x) +1$.

Proof: We start from the left side and then we prove the right side. For the $\cos^2 x$ we use half angle formula, so $$2\cos^2 x \sec(2x)=(1+\cos (2 x))\sec(2x)=\sec(2x)+\sec(2x)\cos (2 x)=\sec(2x) +1$$

We add more examples of the half angle formula for you to practice. To learn about this topic it is really important to try solve these problems by yourself.

Example 3: If $\sin x=\dfrac{2}{3}$ and $x$ is in the first quadrant, find $\sin(\frac{x}{2})$.

Example 4: Find the exact value of $\tan(\dfrac{\pi}{8})$.

Example 5: Find a half angle formula for $\cot x$.

If you need help to understand the trig identities, double angle formulas and half angle formulas, contact us.

Double Angle Formula

In trigonometry, it is really important to know and able to use identities. One of the main and crucial categories of identities is Double Angle Formula. In this post, we talked about different trig identities and formulas. Double Angle Formulas are trigonometric identities that simplify a trigonometric function of $2x$ as of trigonometric functions of $x$. First we recommend to visit the previous post to review identities. Here in this post, first we recall the Double Angle Formula and then we see some examples.

Double Angle Formulas:

Here we have main Double Angle Formulas.

  • $\sin (2 u) =2 \sin u \cos u$.
  • $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$.
  • $\cos (2 u) =2 \cos ^{2} u-1$.
  • $\cos (2 u) =1-2 \sin ^{2} u$.
  • $\tan (2 u) =\dfrac{2 \tan u}{1-\tan ^{2} u}$.

Examples of Double Angle Formulas

Before checking the solutions, try to solve it by yourself first.

Example 1: If $\sin x=\dfrac{3}{5}$ and $x$ is in the first quadrant, find $\tan (2 x)$.

Solution: Since $x$ is in the first quadrant, both $\sin x$ and $\cos x$ are positive. Now we use the Pythagorean identity $\sin^{2} x+\cos^{2} x=1$ to find $\cos x$. From this identity we have, $\dfrac{9}{25} +\cos^{2} x=1$, so $\cos x=\dfrac{4}{5}$. Then $\tan x=\dfrac{\sin x}{\cos x}=\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}=\dfrac{3}{4}$. Now by the last Double Angle Formula, we have $\tan (2 u)=\dfrac{2 \dfrac{3}{4}}{1-\dfrac{9}{16} }=\dfrac{\dfrac{3}{2}}{\dfrac{7}{16}}=\dfrac{48}{14}=\dfrac{24}{7}$.

Example 2: Prove the following identity $\dfrac{1-\cos 2x}{1+\cos 2x}=\tan^2 x$.

Solution: To solve this problem we use the double angle formula for $\cos 2x$. We simplify the left side to get the right side. By the double angle formula we have $\dfrac{1-\cos 2x}{1+\cos 2x}=\dfrac{1-(1-2 \sin ^{2} x)}{1+2 \cos ^{2} x-1}=\dfrac{\sin ^{2} x}{\cos ^{2} x}=\tan^2 x$.

Here we add more examples without solutions for you to practice.

Example 3: Prove the identity: $\dfrac{\cos 2x}{\cos x- \sin x}=\cos x+\sin x$.

The following example is really common in exams.

Example 4: Solve $ \cos(2x)=\cos(x)$ when $0\leq x <2\pi$.

The following example is a bit challenging, but give it a try.

Example 5: Prove the identity: $\cos^4{x}=\dfrac{\cos(4x)}{8}+\dfrac{\cos(2x)}{2}+\dfrac{3}{8}$.

If you need help to understand the trig identities and Double Angle Formulas, contact us.

Prime numbers

What is a prime number?

In mathematics, we say a natural number $p$ is a prime number if p has exactly 2 factors. For example, 11 is a prime number, because the only factors are 1 and 11. It is easy to see that the only prime numbers less than 10 are 2, 3, 5 and 7. The smallest prime number is 2. The only even prime number is also 2. Another equivalent definition of prime number is a natural number greater than 1 that is not a product of two natural numbers greater than 1. For example, 10 is not a prime number, because $ 10=2\times 5$.

What is a composite number?

If a number greater than 1 is not prime, we call it a composite number. The smallest composite number is 4.

Is 1 a prime number?

Note than 1 is not a prime number and it is not a composite number.

List of prime numbers 1 to 100:

In total we have 25 prime numbers less than 100. Here is the list:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

There are a total of 168 prime numbers less than 1000.

How many prime numbers are there?

There are infinitely many prime numbers. There are many proofs for this statement. Let’s prove it here with the most famous method:

On the contrary, let’s assume we have finitely many prime numbers, let’s call them $p_1,…,p_k$. Let $n=p_1\times p_2\times … p_k +1$, it is not difficult to see than $n$ is greater than all $p_{1},…,p_{k}$, so $n$ is not one of them, and then, $n$ is a composite number. So $n$ has a prime factor, let’s say $p_i$ is a factor of $n$. It means than $\dfrac{n}{p_i}$ is an integer, but $\dfrac{n}{p_i}=\dfrac{ p_1\times p_2\times … p_k }{p_i} +\dfrac{1}{p_i}=p_1\times p_2\times p_{i-1}\times p_{i+1}\times … p_k  +\dfrac{1}{p_i}$, which is not an integers, that contradicts our assumption that we have finitely many prime numbers. So, it proves than There are infinitely many prime numbers.

How to check if a number is prime?

To see if a number $n>2$ is a prime number first we check if $n$ is divisible by 2, if it is divisible by 2, then is it not a prime number, then we check divisibility by 3, and then by 5,… . We don’t need to continue to $n$. It is enough to check until $\sqrt{n}$.

For example, to check the number 53, we check this test until $\sqrt{59}$. So we only need to check this test for number, 2, 3, 5 and 7.

What is prime factorization?

Prime Factorization” is factoring our number $n$ into prime factors (prime numbers multiply together to make $n$. For example, the prime factorization of 100 is $100=2^{2}\times 5^{2}$. Prime factors of a number $n$ are prime numbers that are multiplied together to get $n$. As an example, 2 and 5 are prime factors are 10. Note that Prime Factorization is unique.


If you need help to understand the prime numbers, contact us.

Trig Identities

Trigonometry is one of the main topics in math, and many students have issues with trigonometric topics. One of the main topics is trig identities. These trigonometric identities help us with many other topics, including trigonometric equations, derivative with trigonometric functions, integral of trigonometric functions and proving trig identities.

Here we have the list of main trig identities:

Reciprocal identities:

  • $\sin u=\frac{1}{\csc u}$
  • $\cos u=\frac{1}{\sec u}$
  • $\tan u=\frac{1}{\cot u}$
  • $\cot u=\frac{1}{\tan u}$
  • $\csc u=\frac{1}{\sin u}$
  • $\sec u=\frac{1}{\cos u}$

Pythagorean Identities:

  • $\sin^{2} u+\cos^{2} u=1$
  • $1+\tan^{2} u=\sec^{2} u$
  • $1+\cot^{2} u=\csc^{2} u$

Quotient Identities:

  • $\tan u=\frac{\sin u}{\cos u}$
  • $\cot u=\frac{\cos u}{\sin u}$

Co-Function Identities:

  • $\sin \left(\frac{\pi}{2}-u\right)=\cos u$
  • $\cos \left(\frac{\pi}{2}-u\right)=\sin u$
  • $\tan \left(\frac{\pi}{2}-u\right)=\cot u$
  • $\cot \left(\frac{\pi}{2}-u\right)=\tan u$
  • $\csc \left(\frac{\pi}{2}-u\right)=\sec u$
  • $\sec \left(\frac{\pi}{2}-u\right)=\csc u$

Parity Identities: We know that $\sin x$ is an odd function and $\cos x$ is an even function.

  • $\sin (-u)=-\sin u$
  • $\cos (-u)=\cos u$
  • $\tan (-u)=-\tan u$
  • $\cot (-u)=-\cot u$
  • $\csc (-u)=-\csc u$
  • $\sec (-u)=\sec u$

Sum and Difference Formulas:

  • $\sin (u \pm v)=\sin u \cos v \pm \cos u \sin v$
  • $\cos (u \pm v)=\cos u \cos v \mp \sin u \sin v$
  • $\tan (u \pm v)=\frac{\tan u \pm \tan v}{1 \mp \tan u \tan v}$

Double Angle Formulas:

  • $\sin (2 u) =2 \sin u \cos u$
  • $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$
  • $\cos (2 u) =2 \cos ^{2} u-1$
  • $\cos (2 u) =1-2 \sin ^{2} u$
  • $\tan (2 u) =\frac{2 \tan u}{1-\tan ^{2} u}$

To see more examples of Double Angle Formulas visit this page.

Half Angle Formulas:

  • $\sin^{2} u=\frac{1-\cos (2 u)}{2}$
  • $\cos^{2} u=\frac{1+\cos (2 u)}{2}$
  • $\tan^{2} u=\frac{1-\cos (2 u)}{1+\cos (2 u)}$

Sum to product formulas:

  • $\sin u+\sin v=2 \sin \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$
  • $\sin u-\sin v=2 \cos \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)$
  • $\cos u+\cos v=2 \cos \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$
  • $\cos u-\cos v=-2 \sin \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)$

Product to Sum Formulas:

  • $\sin u \sin v=\frac{1}{2}[\cos (u-v)-\cos (u+v)]$
  • $\cos u \cos v=\frac{1}{2}[\cos (u-v)+\cos (u+v)]$
  • $\sin u \cos v=\frac{1}{2}[\sin (u+v)+\sin (u-v)]$
  • $\cos u \sin v=\frac{1}{2}[\sin (u+v)-\sin (u-v)]$

If you need help to understand the trig identities, contact us.

Quadratic formula

What is the quadratic formula?

The quadratic formula is a formula that helps us to find the solutions of an equation of degree 2. First of all, we need to turn our equation of degree 2 to standard form $ax^2+bx+c=0$.

The quadratic formula is: $\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$.

How to use the quadratic formula?

Let’s see some examples of the quadratic formula.

Example 1: Solve the quadratic equation $x^2+5x+6=0$.
Solution: In this quadratic equation a=1, b=5 and c=6. We put these numbers into the quadratic formula and we get:
$\dfrac{-5\pm \sqrt{25-24}}{2}=\dfrac{-5 \pm 1}{2}=-2,-3$.

Example 2: Solve the quadratic equation $x^2+5x+7=0$.
Solution: In this quadratic equation a=1, b=5 and c=7. We put these numbers into the quadratic formula and we get:
$\dfrac{-5\pm \sqrt{25-28}}{2}=$. As you see, under the square root we have a negative number, so there we have no solution for this quadratic equation.

When do we use the quadratic formula?

The quadratic formula is the most general formula to find solutions to quadratic equations. You can use it when you have a quadratic equation.

If you need help to understand and learn about the quadratic formula, contact us.

How to study for math waterloo competitions?

There are many math competitions in waterloo (Pascal, Cayley, Euclid, Fryer, Galois, Hypatia, Gauss, and Fermat). In each of them, you have math questions that you need to answer. Most of the problems are not difficult but still, you need a lot of preparation to be successful. With the math you learn in school, you are not able to solve many problems and get a good result out of these exams. Here are core topics for all of these competitions:

1- Number theory: Gcd, Lcm, Euclid division algorithm, Modulo arithmetic, divisibility, Fermat’s little theorem, Euler theorem, and diophantine equations
2- Combinatorics: Permutation and combination (with repetition problem section), circle computation, binomial theorem, sets, graph theory, recursive counting.
3- Algebra: Polynomials, Quadratic equations, root coefficient relationship, maxima and minima, Solving equations, factoring.
4- Geometry: Solutions of the triangle, Ptolemy theorem, Ceva’s theorem, Menelaus theorem, and other theorems.
5- Problem-solving techniques: Induction, proof by contradiction.

For all of these competitions (Pascal, Cayley, Euclid, Fryer, Galois, Hypatia, Gauss, and Fermat), you need these topics.

Topics in introductory Calculus

Calculus is one of the most important topics in mathematics and usually many students are not good at it. Here there are topics that are so important in calculus that most of the students need to be great at it:

  • Functions (Definition, domain, range, one-to-one, onto and etc)
  • Limits (Definition, one-sided limit, properties, squeeze theorem, continuity and etc)
  • Derivatives (Definitions, slope, product and quotient rule, chain rule and etc)
  • Applications of derivatives (Maxima and minima, first derivative test and etc)
  • Analyzing functions
  • Integrals

We offer tutoring for calculus.

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How to find a math tutor for high school in canada?

Math is one of the most challenging courses in school, especially in high school. It is difficult for many students to learn math. But there are some ways that can make learning math easy and enjoyable. Hiring a qualified and certified math tutor is one of them.

If you seek the best math teacher, we recommend reading this article. Here we offer some tips for how to find a good math tutor.

Before hiring a math tutor, Ask these questions:

There are some important considerations that you should think through before finding a  good math teacher.

● Should your student work with a one-on-one tutor or does he/she can learn math online?

●How much money can you be able to pay for a math tutor?

●How many sessions does your student need to work with a tutor?

●Are there free options to use before hiring a teacher?

There are some places that offer tutoring services free. Do research before paying someone. For example, sometimes your child’s teacher has more time to teach your child before or after school for free.

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Moreover, some local colleges or universities often have free tutoring programs, especially for high school students to help them to prepare for the entrance examination. Ask around and see if there are such programs available.

Another option might be at local community or public libraries. Some local communities often have weekly teaching programs.

If you can’t find any free programs, or they don’t meet your student’s needs, you can try other options: there are lots of teaching websites or videos that can help your student with math. But if you are still looking for someone to help answer your student’s specific math questions, here are some ideas to find a math tutor.

How to find the best math tutor?

There are two ways to find a certified math teacher for high school:

1- Tutoring companies

definitely, Tutoring companies are the best sources to find a qualified teacher. Tutor companies check teacher’s background and their academic qualifications. Therefore, you can confidently choose a skilled math tutor. Moreover, Tutoring companies are almost all over. therefore, You can find a nearer one in your area and save your time!

But tutor companies usually don’t pay the teachers well. So skilled and certified teachers usually have their own business and make more themselves.

2- Private Math Tutors

Hiring a one-on-one tutor is a way to find the best math tutor.  private teachers’ rate is usually higher. However, you get more qualified instruction. In fact, their teaching quality matches their rate.

However, many math tutors are willing to hold their teaching sessions at your home or school and work based on your schedule. This saves your cost and time noticeably. Moreover, the private tutors aren’t limited to fulfilling any other obligations.

So they can teach based on your child’s individual style of learning.

However, hiring a one-on-one math tutor directly has its own difficulties. For example, You yourself have to find a private teacher and check his credential. Decrease the lack of certainty in his background by your own abilities and verifying his certification.

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Where to find a private math tutor?

Finding the best private math tutor on your own may be difficult. But it can help you to save money in a long time as well as get you great outcomes. Here are some sources:

1-Ask Your Friends

The most easiest and obvious approach is asking your friends, neighbors, or your child’s academic counselor to suggest a private math teacher in your area.  Ask them questions like “how much does he/she charge?”  or “How effective is he/she?”

These questions can help you find out whether he is a good fit for your child or not.

2- Check local Bulletin Boards

Local community boards are the other way to find a math private tutor. Many one-on-one tutors advertise their business on such boards. Your local community center, library, bus stops, or metro stations are good places to find such posting. You can also check online boards such as Craigslist or Nextdoor.

If you find a tutor through a posting, it is suggested to have a meeting at a public place such as a public library, then ask him about his background, availability, and cost. You can also ask the following questions:

●Do you have any tutoring experience?

●Which teaching methods do you use to teach math?

●For big tests, will you be available to help more?

●What is your price per hour?

This information might not seem important but they are what you have to base your final decision on while you’re choosing a math tutor for high school. Choose someone who is a good match according to your Priority.

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3-find a tutor online

If you don’t want to ask around for finding a private math tutor, try one-on-one tutoring websites such as schoolmath.ca. These websites are an almost reliable system to communicate with private tutors in different fields. There are lots of qualifies teachers in such online sources. Each teacher has a profile in which you can find every information you need to know about him/her. There are also other students’ comments on that tutor. They help you to decide if a tutor is fit for your child.

Price transparency and the provision of one-on-one and online tutoring services are other advantages of private tutoring websites.

We hope you’ve found this article helpful as you look for the right person or option for your student. What is your idea about how to find the best math tutor for a high school student? If you have any recommendations, please share them in the comments.

If you have any other questions about how we may be able to help you, feel free to text us at 647-249-2491

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Quadratic equations

First of all what is a quadratic equation?

A quadratic equation is an equation that you can write it in the form of $ax^2+bx+c=0$. For example, $x^2+x+5=0 $. Next, how to solve this equation.

There are some methods to solve this type of quadratic equation. One of the main methods is to use the quadratic formula.

QUADRATIC EQUATIONS
Math equations written on a blackboard – mathematics and science concepts

The quadratic formula: $\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$ gives us the solutions. There are some important points:

  • If $b^2-4ac $ is negative, there is no solution.
  • If $b^2-4ac =0$, we have only one solution.
  • If $b^2-4ac $ is positive, we have two solutions.
  • The sum of solutions is: $\dfrac{-b}{a}$
  • The product of solutions is : $\dfrac{c}{a}$.

If you need help to understand the quadratic formula, contact us.

Read More: Linear Equations