I know! Algebra is not used every day. People mock its pointlessness all the time. This showed up on Facebook the other day. You can spend your life escaping algebra! Woo hoo!
What we learned in history, science, and poetry is not useful either. When do you use a historical date? Recite Poe? Balance chemical equations? Calculate acceleration? But, I digress.
I was tutoring a senior (Jay) in precalculus while a junior in the same class (Kay) watched. The following conversation ensued.
Jay: "Mrs. Tammy, when am I ever going to use this?"
Me: "I used this sort of thing to learn high level statistics, and my husband did to learn engineering."
Jay: "I'm not going to major in any of that. This is useless!"
Kay: "My mom teaches math. She never uses this stuff."
So, I admitted what everyone knows. We'll never need to prove trigonometric equations in real life. I shared what I love about these problems. It's like the Gordian's knot. Tug at the right spot, and the whole thing unravels quickly. The same thing that drives Sherlock Holmes to solve mysteries drives me to prove equations. A puzzle. A mystery. A quest. How exciting when you find the right key (or two) and the knots disappear!
Jay had to prove the sides of the equation to be equal even though they look different. Many manipulations shows they are the same. This looks much scarier than it really is!
I ask students what to do first. If I hear a confident "cross multiply," we continue. If I hear a faltering "Cross multiply?" doubt arises. Some students speak jargon that confuses them. So, they launch a buzz word, hoping it's the right one. Then, I ask why cross multiplication. If the explanation makes sense, we move ahead. Otherwise, we linger.
I ask what they can do to make the equation easier. "I don't like fractions!" [That is why we cross multiply.} Then, I ask how to dump the fraction. Most know to multiply the fraction by its denominator (the bottom). I probe further to see if they know why you multiply both sides by the denominator.
The reason so many students falter in math is that they do not understand the why behind what they are doing!
Imagine the left side of the equation sits on the left side of a balance, and the right side on the other. Multiplying the left by a number greater than 1 increases its weight. Doing the same thing to each side keeps the balance. This applet shows this for simple equations! Some pre-algebra materials sow the idea of seeing equations on a physical scale and viewing them as puzzles.
The joy of math is found in unraveling complex mysteries.
Multiplying both sides of the equation by sec θ - 1 scares some students. Working with fractions, however, is scarier! We treat this ugly expression the same as any simple number. Including parenthesis avoids confusion.
We multiply by sec θ - 1 to create sec θ - 1 over sec θ - 1.
Why? Well, what is sec θ - 1 divided by sec θ - 1? Relax! Take deep breaths. What is 2 ÷ 2? One! What is 300 ÷ 300? One! What is a million ÷ a million? One! What is cheese ball ÷ cheese ball? One! So, what is (sec θ - 1) ÷ (sec θ - 1)?
See that's not so hard even if the fraction looked ugly. Now, we have rid ourselves of fractions and all is well.
Some eye the right side of the equation nervously. Why would anyone want to times one number by the difference between two other numbers? Suppose you find your favorite microwave lunch on sale for $2.94 and you want to buy one for each day of the week. You have $20.50 cash on hand. Do you have enough money? While Pamela might be able to multiply 2.94 by 7 in her head, lesser minds like mine can alter the problem slightly and solve mentally, too. The product of 2.94 and 7 is really seven rows of 2.94 as illustrated below.
If we alter 2.94 a tad, we have something easier to manage mentally. What? How can 3.00 - 0.06 be easier than 2.94?
Mulitiplying 3.00 - 0.06 times 7 looks like this.
We can multiply in our head! Yes, 7 times 3 dollars is 21 dollars and 7 times 6 cents is 42 cents. Taking 42 cents away from 21 dollars yields $20.58. Alas, you can only buy six meals.
On the left is another way to illustrate the distributive property. This little rabbit trail has a point for those who do not like the look of cot Θ (sec Θ - 1). You can try to multiply vertically if horizontally worries you.
Sadly, the equation is still ugly. Do you see the next step? Take one step, and the equation looks prettier. Try to focus on what you can remove completely. Think about the analogy of the balance and what you can take off both sides.
Each side of the equation has - cot Θ. Are you stuck? Pretend it's something less scary like - 2. What can you do to - 2 to turn it into zero? Add two! What is - 2 + 2? Zero! What is - 300 + 300? Zero! What is negative million + positive million? Zero! What is negative cheese ball + positive cheese ball? Zero! So, what is negative cot θ + positive cot θ?
ZERO, zip, nada!
The new form of the equation looks almost friendly compared to the original rubbish. At this point in the game (to me, it is a game), I convert everything on the right side to sin θ or cos θ and let the chips fall where they may. These little identities are things that you memorize with use and can always look up if you aren't suffering through a test: csc θ = 1/sin θ and cot θ = cos θ/sin θ and sec θ = 1/cos θ. I simply plug and chug on the right side from here on out!