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# The Sine Ratio

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Using the Sine Ratio in Trigonometry Mathematics

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### The Sine Ratio

1. 1. Image Source: http://seriouslyforreal.com
2. 2. Image Source: http://www.fouman.com When a plane descends it has a speed at an angle, as well as decreasing height, and its flight path forms the hypotenuse of a right triangle. The pilot needs to use the correct angle and speed to make it safely onto the runway at the right location.
3. 3. Opposite Hypotenuse ɵo Opposite-OPP-O ɵSine = OPP HYP ɵSine = O / HɵSin = The “Sine” Ratio for a Right Triangle is defined as the Opposite Side Length divided by the Hypotenuse Length. If we have several right triangles with the same reference Angle, the ratio of their Opposite divided by the Hypotenuse will be the same value for all of these Triangles.
4. 4. Eg. If we look at the Opposite / Hypotenuse for each of the above three Triangles we get Sin37o = 3 / 5 = 6 / 10 = 9 / 15 = 0.6 37 o 37 o 37 o 5 3 10 6 15 9 The Sine Ratio of Opposite / Hypotenuse is the same value.
5. 5. Opposite Hypotenuse ɵo Opposite-OPP-O ɵSine = OPP HYP ɵSin = O / HɵSin = The “Sine” formula can be re-arranged to make a formula for the “Opposite” “O”, by multiplying both sides by “H” OPP HYP ɵSin =HYP x x HYP OPP = HYP x Sinɵ
6. 6. ɵo Opposite-OPP-O OPP HYP ɵSin = The “OPP” formula can be re-arranged to make a formula for the “Hypotenuse” “H”, by dividing both sides by “Sinɵ” Sinɵ OPP = HYP x Sinɵ Sinɵ OPP = HYP x Sinɵ OPP Sinɵ HYP =
7. 7. ɵo Opposite-OPP-O OPP HYP ɵSin = There is a fourth and final Formula for finding the Angle. It is called the “Inverse “Sine” formula. OPP = HYP x Sinɵ ɵ = Sin-1 OPP Sinɵ OPP HYP HYP =
8. 8. ɵo Opposite-OPP-O OPP HYP ɵSin = These are the four formulas for working with Sine Triangles. OPP = HYP x Sinɵ ɵ = Sin-1 OPP Sinɵ OPP HYP HYP = We also use the special “Sin” and “Sin-1 ” calculator buttons when solving Sine Triangles.
9. 9. We use the special “Sin” and “Sin-1 ” calculator buttons when solving Sine Triangle Questions. Warning: Your calculator must be in “Degrees” DEG Mode. Sin 60o sin 60 enter 0.8660 Sin 45o sin 45 enter 0.7071 Sin 30o sin 30 enter 0.5 Note that we round off long decimal trig values from the calculator to four decimal places.
10. 10. To get “Sin-1 ” on the calculator we use “2nd” or “Fn” followed by the “Sin” calculator button. Warning: Your calculator must be in “Degrees” DEG Mode. 60o sin 0.8660 enterSin-1 (0.8660) Note that we usually round off angle values from the calculator to the nearest whole number 2nd 45o sin 0.7071 enterSin-1 (0.7071) 2nd 30o sin 1 enterSin-1 (1/2) 2nd n/d 2
11. 11. 1. Label the Sides of the Triangle 2. Work out if unknown is OPP, HYP, or the Angle. 3a. To find Unknown OPP, use O = H x Sinɵ 3b. To find an Unknown HYP, use H = O / Sinɵ 3c. To find an Unknown Angle, use ɵ = Sin-1 (O / H ) 4. Substitute values into the formula being used 5. Put values into a Calculator and Round Off Answer
12. 12. To find Unknown OPP, use O = H x Sinɵ O = 10 x Sin40 O = 6.42787 O = 6.43 d = 6.43 o d 40 o OPP d 40
13. 13. To find Unknown HYP, use H = O / Sinɵ H = 12 / Sin50 H = 15.66488 H = 15.66 k = 15.66 o 12 50 o 12 50 OPP
14. 14. To find Unknown Angle, use ɵ = Sin-1 (12 / 13) ɵ = 67.38013 ɵ = 67o ɵ ɵ ɵ = Sin-1 (OPP / HYP)
15. 15. To find Sinɵ, use Sinɵ = 12 / 13 Sinɵ = 0.9230769 Sinɵ = 0.9230 ɵ ɵ Sinɵ = OPP / HYP Find Sinɵ