SP#7: Unit Q Concept 2-Finding trig functions using identities

This post was made in collaboration with Jorge Molina. Please visit the other awesome posts on his blog by going here.

Hello there fellow students. Before starting the problem there are a few things you must be aware about. First of all, make sure not to solve this problem straight using trigonometry. We will be using identities for this problem, including: reciprocal identities, ratio identities, and Pythagorean identities. Secondly, solving identities are much easier than we think of them to be. Make sure to look at the clues to help you solve for the problem. Lastly, make sure when rationalizing to be careful what you are multiplying and always look for simplification. Other than that, relax, enjoy and solve your life away!

Photo-credits to the fab-u-lous Victoria Ventura

Photo-credits to the fab-u-lous Victoria Ventura

Photo-credits to the fab-u-lous Victoria Ventura

Photo-credits to the fab-u-lous Jorge Molina

Photo-credits to the fab-u-lous Jorge Molina

Photo-credits to the fab-u-lous Jorge Molina

Photo-credits to the fab-u-lous Victoria Ventura

WPP#12: Unit O Concept 10-Solving angle of elevation and depression word problems

Pooky's great adventure in Disneyland!:

One beautiful Tuesday afternoon, while roaming around looking for her parents, Pooky stumbled upon the Sleeping Beauty Castle at Disneyland. She was admiring its beauty when all of the sudden she was kidnapped by the troublesome Tinkerbell! GASP. (Here's the deets: Tinkerbell was jealous that as Peter Pan flew past the castle, he admiringly stared at Pooky.) Unfortunately, Tinkerbell placed Pooky in the highest tower, making it difficult for Pooky to be rescued.

While Pooky waited for Prince Charming to save her from the tower, she decided to measure Matterhorn's mountain height. Being the smart-alec she was, she figured out something incredible.

She measured the angle of elevation to the mountain across to be 29 degrees and the angle of depression (to the base of the mountain) to be 40 degrees. If the two, the castle and mountain, are 75 feet apart, how tall is the Matterhorn Mountain? (Round to the nearest foot) 

Here's a lovely picture of Pooky admiring Disneyland:

Picture made and taken by the lovely Victoria Ventura

Here's the real business and work for the problem:

Picture made and taken by the lovely Victoria Ventura

ANY QUESTIONS? FEEL FREE TO COMMENT.

I/D#2: Unit O-How can we derive the patterns for our special right triangles?

Inquiry Activity Summary:

30-60-90

Credits: To the wonderful and beautiful Victoria Ventura

So you may be asking yourself, "what is this hoopla that I am looking at?" Its not as hard as you think or at least how it looks. Let me simplify it for you. The triangle is an equilateral triangle all the sides equal 1. But this means all angles are 60 degrees? So what do I do? You split them in half, making it into two 30-60-90 degree triangles. We know for a fact from our unit circle the bottom "x" side would be 1/2 and the "y" side would be radical3/2. And the hypotenuse would be one.

Now how could this be applied to all different triangles that simply don't have the hypotenuse as one? We'd simply add a variable to make it applicable to all side lengths possible ever! This is the pattern if you haven't caught up...

Below is a picture of an example to help you further on your understanding of 30-60-90 triangles:

Credits: To the wonderful and beautiful Victoria Ventura

45-45-90

Credits: To the wonderful and beautiful Victoria Ventura

Much simpler than the 30-60-90 triangle is the 45-45-90 triangles. Just as we did before with the the equilateral triangle, we would have to split the square in half. Splitting directly through two 90 degree angles. Thus, you'll get two 45-45-90 triangles. Fabulous, I know. All sides are equal to one. But what would the hypotenuse equal? This is where our Pythagorean theorem would take play. 1^2 + 1^2=2 which in fact would be radical2. To make it relate-able to other 45-45-90 triangles, just as we did with 30-60-90 triangles, we would multiply all sides with n.

With this added variable, it would be a great pattern to follow and find other missing sides, knowing limited information.

Here's a fabulous example further showing solving 45-45-90 triangles having only limited information!:

Credits: To the wonderful and beautiful Victoria Ventura

Inquiry Activity Summary:

Something I never noticed before about special right triangles is how simple they can actually be. I made my life misery last year I over-thought things and kind of just got tired so I memorized the sides. I never put thought into how the sides were derived. This I/D really helped me to see that in a clearer light and understand how these triangles actually got the sides.

Being able to derive these patterns myself aids in my learning because I'm learning to be able to solve things on my own without help of anyone else. Of course, just like any other human on planet earth, I get a little help from my friends from time to time but, only when most needed. Just like the previous I/D, it really helps students to learn to actually practice thinking, not just doing careless work and understanding why, what and how and a problem can be solved, such like this I/D. It really helps us see our strengths and weaknesses and is a major help to students.