This week in PreCalc 11, we started to learn about Trigonometry. We learned many concepts regarding trigonometry, but the one thing that stood out to me was the Sine Law. The reason why it stood out to me was because it cuts down the work we had to do last year to solve the same things that the Sine Law does. Last year if we were introduced to triangles that weren’t typical 90 degrees angles, we had to make a line from the top point perpendicular to the bottom line to solve each angle and solve for each sides measurements. Fortunately, the Sine Law helps us do that with less amount of work needed to be done.

The Sine Law is a/sin A = b/sin B = c/sin C, where the lower case letters are the measurements of the side lengths and and the upper case letters are the lower case letters corresponding angle. By corresponding angle i mean that if angle B describing one point of the triangle, then side b would be the side of the triangle that is directly opposite of angle B. Because the Sine Law works with the angles of triangles and the measurements of the triangles side then it’s useful in finding a missing angle or side. 

Example: 

Work:

Step 1: draw triangle

When given a problem that describes the angles and the side lengths of a particular triangle, the first step is to always draw it. Drawing the triangle gives us a better understanding of what we have and what we’re looking for with little room for error. In this example, we we’re given two side lengths, KN and MK , and angle M. I drew the triangle, labelled the points, and then labelled the characteristic given. That’s when I found out what I was looking for, angle N, angle K, and side MN.

Step 2: find the equation to use first

The easiest way to do this is to label the sides first. In this example, opposite of angle K would be side k, opposite angle M would be side m, and opposite angle N would be side n. Once we have labelled those we should write out the full equation and plug in everything we know. The equation comes with three different ones, but we can only use two at a time. So the important thing is to use two equations that we can solve for one variable meaning all other variables we should already have. In this example, we have Sin M, m, and n, meaning we could find Sin N.

Step 3: solve the equation

Once we have the first equation we are going to use to solve for something, then all we have to do is solve for it. To solve for it, I used cross-multiplication and solved from there. When i moved Sin to the other side, I inverted it.

Step 4: find the last angle

Once we found angle M, we can easily find the last unknown angle with little work. We know that the 3 angles of one triangle have to add up to 180 degrees. So we know that one angle is 70 degrees and another angle is 75, so that means if we take 180 and subtract 70 and subtract another 75, it will give us the final angle. In this case, the final angle is 35 degrees. That means we have angle M as 70 degrees, angle N as 75 degrees, and angle K as 35 degrees.

Step 5: solve for the last side length

The only thing we have left to solve is the final side length. We can use the Sine Law to find the last side length. Since we have all variables of the equations except for k then we can use any two of the equations to find it. The final step is to solve for it. In this example the final side length was 8.6 cm.

Now we know all the triangles angles and side lengths all thanks to the Sine Law!

This week in PreCalc 11 we learned how to solve rational equations. Rational equations are equations containing at least one fraction whose numerator or denominator is a variable.

There are two ways to solve rational equations, one of them is multiplying every term by each of the denominators or cross multiplication. Cross multiplication only works when there are two fractions and one is on each side of the equal sign whereas multiplying by the denominator is a strategy that will work with every type of rational equation.

Example:

Explanation:

Step 1: Factor

Factor any polynomials that aren’t already factored. To factor, find two numbers that when multiplied together equal the last term and when added together equal the middle term.

Step 2: Multiply by the Denominator

Multiplying by the denominator is just doing the opposite operation to take the fraction away to make the equation easier to solve. When you have a fraction it means that its the numerator divided by the denominator, so to get rid of the denominator, we should multiply the term by the denominator to get rid of it. If we multiply the term by it, then the same term that is on the denominator will also be apart of the numerator making them cancel out. But what we do to one term, we do to every term, so we have to multiply each term in the equation by the denominator. Typically there is more than one denominator, so we follow the same procedure for the rest of the fractions.

Step 3: Non-Permissible Values

Non permissible values are numbers in which x cannot equal because it will make the denominators zero which can never happen. So stating non permissible values is important. In this case, the denominators consists of (b – 2), (b +2), and (b – 3) so b cannot equal 2, -2, and 3 because it will make the denominators equal zero.

Step 4: Solve

The final step is to solve. Multiply each term by the denominators, cancel out similar factors that are on the numerator and denominator and solve for x.