During this week of Pre-Calculus 11, we reviewed trigonometry equations that included special triangles and the unit circle.

Key Vocab:

Unit Circle – A unit circle on an x-y coordinate plane where the center of the unit circle is at the origin and the circumference of the circle touches

Special Triangles – 30-60-90, 45-45-90. These triangles are special triangles because the ratio of their sides is known to us so we can make use of this information to help us in right triangles trigonometry problems.

Conterminal Angle – are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side.

When we first talked about the special triangles, we were looking at the angles and the sides of triangles of 45-45-90 and 30-60-90 degrees triangles.

The Unit Circle was also a key factor to know and memorize as it will help determine whether sin, cos, or tan can equal 0, 1, or -1.

The reason why they are called special is that with simple geometry and solving, we can figure out the ratios of their sides and easily solve the problem without any use of a calculator.

For example 1, we can see that the degree is equal to 1/2 of cos. Without using a calculator, we can use the triangle that has a side of 1 and 2. Which comes to the 30-60-90 triangle. Then we figure which angle will make the 1 side adjacent to the degree. Which then would be 60 degrees. But remember that there are always more than one answer, most likely 2 possible answers.

To find the 2nd answer, we will need to figure out what quadrant they are in. If it is positive, we need to find the quadrant that cos would be positive. You can use the CAST strategy. So the quadrants that would be positive would be Q1 and Q4. The reference angle would be 60 degrees and you would need to find the conterminal angles for Q1 and Q4.

So the two answers should be the conterminal answers 60, and 300 degrees.

In the second question, we have tan degrees would = 0. First of all, there are no triangles that have 0 as one of its sides. So we would need to use the Unit Circle instead for this problem.

To use the Unit Circle, we need to remember that tan = y/x , sin = y/R, and cos =  x/R and whereas R=1. Since we are using tan, we need x and y. So looking at the coordinates, we need y = 0 and x to equal any number, since anything divided by 0=0. So the two possible answers would be (1,0) and (-1,0). These two coordinates are located at 360 degrees and at 180 degrees. Those are the final answers to this expression.

This is really important to remember because it is something that you can easily remember and quickly write out to help you find a degree or a side without doing any calculations with a calculator.

During this week of math, we went through and reviewed trigonometry from grade 10 to have a sense of what we are going to be doing in Unit 5.

Key Vocabs:

Pythagorean theorem – the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse

Sine, cosine, tangent – Sin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles

In the first diagram/question, we can see that we are given two values of the sides of the triangle. We can easily determine the missing side length using the Pythagorean theorem.

Then on the right, we have come to a more difficult problem. We have the length of one side and the degree of one angle. Our goal is to find the hypotenuse from this information.

We can use the sine, cosine, and tangent signs we have learned in grade 10 to figure out the missing side length.

On this example, we can see that we need to find the degree for the angle instead of finding a missing side length. But with only the information of 2 side lengths, we need to use either sine, cosine, or tangent. Since we have an opposite and an adjacent, we can use tagine to figure it out.

Since we just want to isolate the degree sign alone, we have to move the tangent over so it becomes a tangent-1 and it will multiply with the fraction on the other side as well. Then put it in your calculator and your answer should be good to go. Make sure to round your answer to the nearest degree (whole number).

This was important to know because it will greatly help in future trigonometry questions in Unit 5 and onwards. It was a great time to review it as well since we might have forgotten how to solve them in the first place.

During this week of Pre-calculus 11, we continued on unit 6 with how to solve rational equations.

Key Vocab:

Quadratic – something that pertains to squares, to the operation of squaring, to terms of the second degree or equations or formulas that involve such terms

One of the first steps we went over was what happens when you have a fraction that equals another fraction?

For an example.

If you have a fraction that equals to another fraction but has a binomial at the bottom, we would want to cross-multiply them.

this cross-multiplying only works if you have an equal sign between the fractions. If you don’t, then the cross-multiplying will not work.

After you have cross-multiplied them, you can then figure out what m is.

Also make sure to check what the non-permissible values for the equation is because if your answer if one of the non-permissible values, then there was an error mistake in your work or either the question.

For the second example, we have almost the exact same question as in the first example. But this time, when we cross multiplied them, we get a quadratic equation.

If in this case we end up with a quadratic equation, we will have to factor it out at the end.

Using the three methods of factoring, you can use their to factor. I chose to use the quadratic formula.

After factoring, if you can not simply it anymore, then that is your final answer.

Always remember to put in you non-permissible values as well. As that will help you to figure out if your answer is correct or not.

This is important to know because in most cases we forget that we can cross multiply whenever we have an equation of a number equals to another number. Cross-multiplying is always something we can use daily in math and precalculus and an important strategy to keep in mind whenever you are solving these types of equations.