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Introduction: Exams can be a daunting aspect of any course, and Math 135 at the University of Waterloo is no exception. However, with the right preparation and guidance, you can confidently approach Math 135 exams and achieve outstanding results. At schoolmath, we offer expert tutoring services specifically tailored to help you excel in Math 135 exams. In this article, we will discuss the importance of exam preparation, highlight key strategies for success, and showcase how our tutors can support you in achieving your academic goals.

1. Understanding the Significance of Exam Preparation: Math 135 exams at the University of Waterloo are designed to assess your comprehension of the course material and your ability to apply mathematical concepts effectively. Adequate exam preparation is crucial for achieving the best possible outcomes. Our tutors recognize the importance of exam readiness and will guide you through comprehensive preparation strategies to maximize your performance.
2. Comprehensive Review of Course Material: Our tutoring services provide you with a thorough review of the Math 135 curriculum, ensuring that you have a solid understanding of all key concepts covered in the course. Our expert tutors will identify any knowledge gaps you may have and help you fill them through targeted lessons, practice problems, and concept reinforcement. This comprehensive review prepares you for the breadth and depth of topics that may appear on your exams.
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Conclusion: Math 135 exams at the University of Waterloo may pose challenges, but with expert tutoring support from Schoolmath, you can overcome them and achieve exceptional results. Our tutors specialize in preparing students for Math 135 exams, providing comprehensive reviews, effective study techniques, ample practice opportunities, and valuable test-taking strategies. With our guidance, you will gain confidence, develop strong problem-solving skills, and maximize your performance in Math 135 exams. Let us unlock your full potential and help you achieve the academic success you deserve. Contact us today and embark on your journey towards acing Math 135 exams!

Calculus is a branch of mathematics that deals with rates of change and continuity. It is a fundamental tool in many areas of science, engineering, and economics. To be successful in calculus, there are a few key things you can do to help you understand the material and excel in your studies.

Start with a solid understanding of the basics: Before diving into calculus, make sure you have a good grasp of basic algebra and pre-calculus concepts such as functions, trigonometry, and exponential and logarithmic functions. Without a solid foundation, it will be difficult to understand the more advanced concepts in calculus.

Practice, practice, practice: As with most subjects, practice is key to success in calculus. Make sure you are working through plenty of problems and exercises to help solidify your understanding of the material. Try to work on problems that are similar to the ones you will encounter on your exams.

Understand the concepts, not just the formulas: Calculus is more than just a set of formulas to memorize; it's a way of understanding how the world around us works. Try to understand the concepts behind the formulas rather than just memorizing them. This will help you apply the concepts to new situations and make connections to other areas of mathematics and science.

Use visualization and geometric intuition: Calculus is a visual subject and using visualization can help you to understand the concepts better. Try to visualize the concepts as much as possible and to understand how the graphs of functions change as you vary their parameters.

Stay organized and take good notes: keep your notes in an organized fashion and maintain a clear record of the main theorems, results, and formulas.

Stay positive and avoid procrastination: Stay positive, don't give up easily, and avoid procrastination. It's important to stay on top of the material and not to fall behind as it makes it much harder to catch up.

Remember the big picture: Remember the big picture and how the material you're learning fits into it. Knowing the big picture will give you a better understanding of the material, and it will also help you understand the importance of calculus in many areas of science and engineering.

Seek out additional help when needed: If you find yourself struggling with the material, don't hesitate to seek out additional help. Talk to your instructor or TA, attend office hours, or consider working with a tutor. If you need to work with a tutor we can help you. Please contact us.

These are some of the key strategies that can help you be successful in calculus. Remember that success in calculus requires dedication, hard work, and perseverance. Stay motivated, stay organized, and always strive to understand the concepts behind the formulas. With the right mindset and approach, you can excel in your calculus studies.

MATH 135 (Algebra for Honours Mathematics) is one of the most difficult first-year courses at the 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, the 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 the most important part of Math 135.

If you need help with Math 135, contact us.

Trig identities are a crucial part of trig functions, and one of the most important identities in trigonometric functions is the half-angle formula. In this post, we discussed various trigonometric identities and formulas. First, we recommend visiting the previous posts to review identities, including the double-angle formula. Here we talk about half-angle formulas. We introduce them, then we demonstrate their use with some examples and applications.

What is the half-angle formula?

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

• 
• 
• 

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

Examples of half angle formula:

Example 1: Find .

Solution: To solve this problem, we use the half angle formula for the function $\sin$.
$$
We know that . So $$
Since is in the first quadrant, then $latex\sin (\dfrac{\pi}{12})$ is positive, so
$$

Example 2: Prove the identity .

Proof: We start from the left side, and then we prove the right side. For the we use half angle formula, so $$

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

Example 3: If and $latexx$ is in the first quadrant, find .

Example 4: Find the exact value of .

Example 5: Find a half-angle formula for .

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

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.

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.

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.

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.

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.

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.