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# The Cosine Ratio

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The Cosine Ratio in Trigonometry Mathematics

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### The Cosine Ratio

1. 1. Image Source: http://rekkerd.org
2. 2. Image Source: www.rkm.com.au - used with permission. The graphs of the values of Sine and Cosine from Right Triangles create shapes that are very much the same as sound waves. We can electronically make these sound wave shapes and then alter their shape and sound with guitar pedals.
3. 3. Image Source: Google Images. Distorted Heavy Metal guitar sound occurs when smooth Sine and Cosine Waves are mathematically transformed into Square Shaped, Sawtooth, or Triangle shaped waves.
4. 4. Adjacent Hypotenuse ɵo Adjacent - ADJ - A ɵCosine = ADJ HYP ɵCosine = A / HɵCos = The “Cosine” Ratio for a Right Triangle is defined as the Adjacent Side Length divided by the Hypotenuse Length. If we have several right triangles with the same Reference Angle, the ratio of their Adjacent divided by the Hypotenuse will be the same value for all of these Triangles.
5. 5. Eg. If we look at the Adjacent / Hypotenuse for each of the above three Triangles we get Cos37o = 4 / 5 = 8 / 10 = 12 / 15 = 0.8 37 o 37 o 37 o 5 4 10 8 15 12 The Cosine Ratio of Adjacent / Hypotenuse is the same value.
6. 6. ɵo ADJ HYP ɵCos = These are the four formulas for working with Cosine Triangles. ADJ = HYP x Cosɵ ɵ = Cos-1 ADJ Cosɵ ADJ HYP HYP = We also use the special “Cos” and “Cos-1 ” calculator buttons when solving Cosine Triangles. Adjacent - ADJ - A
7. 7. We use the special “Cos” and “Cos-1 ” calculator buttons when solving Cosine Triangle Questions. Warning: Your calculator must be in “Degrees” DEG Mode. Cos 60o cos 60 enter 0.5 Cos 45o cos 45 enter 0.7071 Cos 30o cos 30 enter 0.8660 Note that we round off long decimal trig values from the calculator to four decimal places.
8. 8. To get “Cos-1 ” on the calculator we use “2nd” or “Fn” followed by the “Cos” calculator button. Warning: Your calculator must be in “Degrees” DEG Mode. 30o cos 0.8660 enterCos-1 (0.8660) Note that we usually round off angle values from the calculator to the nearest whole number 2nd 45o cos 0.7071 enterCos-1 (0.7071) 2nd 60o cos 1 enterCos-1 (1/2) 2nd n/d 2
9. 9. 1. Label the Sides of the Triangle 2. Work out if unknown is ADJ, HYP, or the Angle. 3a. To find Unknown ADJ, use A = H x Cosɵ 3b. To find an Unknown HYP, use H = A / Cosɵ 3c. To find an Unknown Angle, use ɵ = Cos-1 (A / H ) 4. Substitute values into the formula being used 5. Put values into a Calculator and Round Off Answer
10. 10. To find Unknown ADJ, use A = H x Cosɵ A = 10 x Cos35 A = 8.19520 A = 8.20 k = 8.20 o k 35 o ADJ k 35
11. 11. To find Unknown HYP, use H = A / Cosɵ H = 12 / Cos40 H = 15.66488 H = 15.66 m = 15.66 o 12 40 o 12 40 ADJ
12. 12. To find Unknown Angle, use β = Cos-1 (12 / 13) β = 22.61986 β = 23o β ɵ = Cos-1 (ADJ / HYP) β
13. 13. ββ To find Cosɵ, use Cosβ = 12 / 13 Cosβ = 0.9230769 Cosβ = 0.9230 Cosɵ = ADJ / HYP Find Cosβ
14. 14. http://passyworldofmathematics.com/ All slides are exclusive Copyright of Passy’s World of Mathematics Visit our site for Free Mathematics PowerPoints