I/D#3: Unit Q Concept 1: Using Fundamental Identites to Simplify & Verify Expressions

Inquiry Activity Summary

"sin²x+cos²x=1" comes from the identity (proven factor or formula) of the Pythagorean Theorem. The theorem, when involving the trig ratios of the Unit Circle, is written in the form of "x²+y²=r²". In order to get the set of values to equal "1", we must divide the entire equation by "r²". As a result, we are left with two ratios, "x/r" (cosine), and "y/r" (sine).

Inquiry Activity Reflection

The connections that I see between Units N, O, P, and Q so far are the use of angels in degrees and the use of geometric properties. Likewise, I have noticed that all of the units make use of the six central trigonometric ratios.

If I had to describe trigonometry in THREE words, they would be: patterns, formulas, and memorization.

I/D#2: Unit O: How Can We Derive the Patterns for Special Right Triangles?

Inquiry Activity Summary

The objective of the activity below was to derive or obtain the patterns of special right triangles from equilateral shapes, in this case a larger triangle and a square. The benefits of completing this task is that it provides us with another means of identifying the Unit Circle values.

Activity page 1 of 2. Click to enlarge.

Activity page 2 of 2. Click to enlarge.

45°, 45°, 90° Triangle

http://www.gradeamathhelp.com/image-files/45-45-90-triangle.jpg

           As shown in the Step 1 of the above activity (see page 1), the process of deriving a 45°, °45, 90° congruent triangles of the desired angles. We label the hypotenuse of the smaller triangles "c", and we can label the two legs "a" and "b". Based on what we informed in this activity, we know that the side lengths of the square were valued at 1; thus, both "a" and "b" are equal to 1. To find the value of the hypotenuse, we can use the Pythagorean Theorem (a² + b² = c²). By plugging in the corresponding values, we get c= √2. We can then collectively use the newly-obtained values and a variable (e.g. "n") to triangle from a square first involves the act of dividing, or breaking down, the initial shape. To do so, we slice the square diagonally to from two form a general pattern. This means that by adding a variable to each of the values, the pattern becomes applicable to a square with any given side length!

30°, 60°, 90° Triangle

http://www.gradeamathhelp.com/image-files/30-60-90-triangle.jpg

           In Step 2 of the activity (see page 2), we are presented with an equilateral triangle with a side length of 1. Referring back to what we have learned in Geometry, we know that a triangle with sides of equal length is also equiangular, meaning that each of the three angles are 60°. To begin this step, me must again breakdown the shape by dividing it into congruent parts--this time by drawing a vertical line through the middle. Again we label out hypotenuse "c" and the two legs of the shape "a" and "b". Using the given information, we know that "a" is equal to 1/2 and "c" is equal to 1. As you may have probably already guessed from Step 1, we use the Pythagorean Theorem to find the missing height. By plugging in the respective values into the equation and solving for "b", we get the answer of √3 /2. In order to translate our values into the normal pattern without fractions, we multiply each of them by 2. By doing so, "a" becomes 1, "b" becomes √3, and "c" becomes 2. As stated before, we can include a variable (e.g. "n") to our answers to form a general pattern. This allows the pattern to be applicable to an equilateral triangle with any given side length!

Inquiry Activity Reflection

Something I never noticed before about special right triangle is that they can be joined together to form larger, equilateral and equiangular shapes!

Being able to derive these patterns myself aids my learning because it enables me to identify patterns that could prove useful in future practice and exams.

Works Cited

• http://www.gradeamathhelp.com/image-files/45-45-90-triangle.jpg
• http://www.gradeamathhelp.com/image-files/30-60-90-triangle.jpg 

I/D#1: Unit N: How Do Special Right Triangles and the Unit Circle Relate?

Inquiry Activity Summary

http://gmatprepster.com/wp-content/uploads/2012/07/GMAT-Right-Triangles-1.png

            The objective of the below in-class activity was to derive the values of the Unit Circle from two types of special right triangles (right). For each of the three examples, we labeled the shapes according to the rules of Special Right Triangles, simplified the values of the sides, and identified the three vertices of each as ordered pairs.

Activity page 1 of 2. Click to enlarge.

Activity page 2 of 2. Click to enlarge.

 

 

 

 

 

 

 

 

 

 

30° Triangle

http://www.montereyinstitute.org
/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U19_L1_T3
_text_final_3_files/image099.gif

      As seen in the first example of the activity, the shape contains three angles (30°, 60°, 90°). Thus, according to the rules of Special Right Triangles, the hypotenuse (r) is valued at 2x; the horizontal value (x) is x; the vertical value (y) is x√3. Due to the fact that we are instructed to simplify so that the "r" is equal to 1, we divide 2x by 2x. Just as in any given equation, what we do to one side, we must do to the other; we divide the values of the other two sides of the triangle by 2x. We are left with "r"= 1, "x"= √3/2, and "y"= 1/2.

 

 

 

 

 

 

45° Triangle

http://www.montereyinstitute.org
/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U19_L1_T3
_text_final_3_files/image036.gif

          As seen in the second example of the activity, the shape contains three angles (45°, 45°, 90°). Therefore, according to the rules of Special Right Triangles, the hypotenuse (r) is valued at x√2; the horizontal value (x) is x; the vertical value (y) is also x. As before, we simplify so that the "r" is equal to 1, meaning we divide x√2 by itself. Afterwards, we divide the values of the other two sides of the triangle by x√2. For the "x" and "y" values, we are left with an unacceptable radical in the denominator. To resolve this, we multiply both the top and the bottom by √2 and simplify.In the end we get "r"= 1, "x"= √2/2, and "y"= √2/2. Using this information, we then find the coordinates of the vertices that can be seen on page 2 of the activity.

 

 

60° Triangle 

http://www.montereyinstitute.org
/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U19_L1_T3
_text_final_3_files/image108.gif

Similar to the first example, the third shape has three angles (60°, 30°, 90°). Thus, according to the rules of Special Right Triangles, the hypotenuse (r) is valued at 2x; the horizontal value (x) is x; the vertical value (y) is x√3. We are instructed to simplify so that "r" is equal to 1, so we divide 2x by 2x. Like before, we must divide the other two sides of the triangle by 2x. We are left with "r"= 1, "x"= 1/2, and "y"= √3/2. Using this information, we find the vertices of the shape on a coordinate plane (see page 2 of activity).

  

 

 

 

http://www.pccmathuyekawa.com/classes-taught/math_7ab/unit%20circle.jpg

          

       
           By labeling and simplifying the values of the special right triangles, I was able to identify the vertices of the shapes. If I were to plot these points within the first quadrant of a coordinate plane, I would obtain one fourth of the Unit Circle! Using this information, I could simply flip the triangles across the axes of the said coordinate plane as depicted (right) to find the rest of the circle.

           

          The triangles drawn in this activity lie within the first quadrant of a coordinate plane, where both the "x" and the "y" values of the vertex are positive. As shown in the picture to the left, if a triangle was flipped across the x-axis (quadrant IV), the x-value would remain positive, while the y-value would become negative. Conversely, if a triangle was flipped across the y-axis (quadrant II), the x-value would become negative while the y-value would remain positive. If a triangle was flipped both across both axes (quadrant III), then both the "x" and "y" values would become negative.

 

Inquiry Activity Reflection 

The coolest thing I learned from this activity was that when memorizing the Unit Circle, I only actually have to recall one fourth of it! This activity will help me in this unit because it will help me remember the values and degrees of the unit circle in preparation for the upcoming test. Something I never realized before about special right triangles and the unit circle is that when the shapes are drawn on a coordinate plane, their vertices can be connected to form a circle!

To learn more about angles and the unit circle, visit Crystal Kirch's blog here!

Works Cited 

• http://gmatprepster.com/wp-content/uploads/2012/07/GMAT-Right-Triangles-1.png 
• http://www.pccmathuyekawa.com/classes-taught/math_7ab/unit%20circle.jpg
• http://kirchmathanalysis.blogspot.com/p/unit-n.html
•  http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U19_L1_T3_text_final_3_files/image099.gif
• http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U19_L1_T3_text_final_3_files/image036.gif 
• http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U19_L1_T3_text_final_3_files/image108.gif