I/D#3: unit Q-Pythagorean Identities

• First of all an identity is a proven fact that is always true. The Pythagorean Theorem when using x,y and r, it's x^2+y^2=r^2. Then divide everything by r^2 to get one. It will look like this (x^2/r^2)+(y^2/r^2)=1. Since cosine is x/r and sine is y/r. So it will be replace to be cos^2x+sin^2x=1. That's why it's referred to the Pythagorean Theorem. The magic 5 pairs to make an identity true are (1/2,3/2),(2/2,2/2),(3/2,1/2), (0,1) and (1,0). The connection between units N, O, P, and Q so far are the magic 5 pairs or the unit circle and using sin, cos and tan as well as its reciprocals. If I had to describe trigonometry in THREE words, they would be complicated, long and strategically to solve. 
• 
  
• For further understanding visit this websites, number three is a video explaining the process:
• 1.http://math.tutorvista.com/geometry/pythagorean-identities.html?view=simple
• 
  
• 2.http://www.purplemath.com/modules/idents.htm
• 
  
• 3.http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-27-trig-integrals/

WPP#13&14:unit P concept 6&7

Please see my WPP13-14, made in collaboration with Jorge Barroso, by visiting his blog http://jorgebperiod1.blogspot.com/2014/03/wpp-13-14-unit-p-concepts-6-7.html.  Also be sure to check out the other awesome posts on his blog. 

Unit P concept 3-5

3. Law of Cosines
Why do we need it? How is it derived from what we already know?

We need law of cosines to find the third side when knowing two sides and an angle (SSA) or knowing all three sides (SSS) to find the angles.

We got a triangle label ABC that's not a right angle. Angle A's point is (0,0), angle B's point is (ccosA,csinA), ccosA is the x value because cosA equals d/c and csinA is the y value because sinA equals h/c. Angle C's point is (b,0), b is the distance across from angle B between angle A and C. Draw a line down angle B to make h, h is the drawn height of the triangle. With that it makes a 90 degrees angle. Then label side c and d also ccosA. Side c is the distance across from angle C between angle A and B. Side d is the distance across from angle B in just the right triangle. Side a is the distance between angle B and C (see picture 1). Use the Pythagorean Theorem in triangle CBD but substitute h to csinA and r to ccosA. Foil to get b^2-2bccosA+c^2cos^2A (see picture 2). Put cos^2A + sin^2A in parenthesis, since since cos^2A + sin^2A equals one. Finally get the formula (see picture 3), it can also be used to produce other letters statements (see picture 4). That's how we derive law of cosines.

Picture 1

Picture 2

Picture 3

Picture 4

Refrence:
http://www.regentsprep.org/Regents/math/algtrig/ATT12/derivelawofsines.htm

5. Area formulas
 Draw out a right triangle with angle A equal to 35 and angle B equal to 65. Side b equal to 4. Use the law of sines to get side b and c. Use the area of an oblique triangle, and Heron’s area formula to get both areas (see picture 5).

Picture 5

 Work cited

http://www.regentsprep.org/Regents/math/algtrig/ATT12/derivelawofsines.htm

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

A. Juanito at ground level measures the angle of elevation to the top of the cliff is 44 degrees. If he is 7 feet away from the cliff, what is the height of the cliff?

B. Junito climbs to the top of cliff and wants to glide down the cliff. The angle of depression is 47 degrees. The landing site is 35 feet apart, how tall is the from that side?

Remember opposite, adjacent, and hypotenuse and SOHCAHTOA.

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

For the 30-60-30 triangle label all the sides by 1. Divide the triangle in two. That will divide the 60 degrees in to 30 degrees, the adjacent in half and make a 90 degree. Use the Pythagorean theorem to get the opposite side. Multiply by 2 to leave the radical alone and get 1 in the adjacent side and get 2 in the hypotenuse side. 

 For a 45-45-90 triangle label all the sides by 1. Divide the square to get two triangles. That will divide the 90 degrees into 45 degrees. Use the the Pythagorean theorem to get the hypotenuse side.  

 We use "n" to reference the missing value.

Something I never noticed before about special right triangles is, why we use "n" for?

Being able to derive these patterns myself aids in my learning because it gives a picture of both triangles and its formula. 

This are pictures of both triangles and showing how to get their formulas.

30-60-30 triangle

45-45-90 triangle