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Introduction to trigonometry M. S. Martin Feb 2007

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Basic trigonometry equations SIN = OPP HYP COS =ADJ HYP TAN =OPP ADJ Remember SOH, CAH, TOA deg° OPPOSITE ADJACENT HYPOTENUSE

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Now lets look at the calculator Find these functions on the calculator. Sin, Cos, Tan

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Now lets look at how we use the equations We will use an example of 28° and the right angle triangle opposite We know the degrees, we know the length of one side, but not the length of the opposite side ? So we can apply the ‘tan’ formula here Tan 28° = OPP or ? ADJ 3.6 We have to re-arrange the formula OPP = Tan 28° x 3.600 Tan 28° x 3.600 = 1.914 Now lets just look at that math's HYPOTENUSE OPPOSITE ADJACENT 28° 3.600m ?

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Re-arrangement of the equation Tan 28° = OPP or ? ADJ 3.6 Same as, Tan 28° = OPP 1 3.600 Apply cross multiplication 1 x opp = tan 28° x 3.600 opp = tan 28° x 3.600 = 1.914 An easy way to remember for rearranging other formula’s is Think of a simple equation 4 = 8 2 Find the corresponding number to your formula and look at what you have to do to the other numbers Meaning, you want 8 so 4 x 2 = 8 If you wanted 2, 8 ÷ 4 = 2 X

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Write the equation for the x° in the following triangles X°X° 6 9 8 Tan x = Sin x = Cos x = 686989686989 X°X° Tan x = Sin x = Cos x = 5 8 7 575878575878

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Write the equation, then solve it 30° 50m ? Sin 30° = OPP HYP Sin 30° = OPP 50 Re-arrange OPP = Sin 30° x 50 = 25 25° ? 14.670m Cos 25° = ADJ HYP Cos 25° = 14.670 HYP Re – arrange HYP = 14.670 Cos 25° = 16.187

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Now let’s look at how it relates to a building Firstly to calculate the drop off measurement in a roof frame

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Drop off measurements Can you see the drop off measurement ? Can you see the relevent triangle ? Handout, Now lets look at a example

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Use ‘Tan’ to calculate drop off Tan 30° = opposite side adjacent side Tan 30° = opposite side 400 + 110 + 40 Tan 30° = opposite side 550 Re-arrange formula Opp side = tan 30° x 550 = 317mm Total drop off = 317 + 20 = 337mm

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Pitch and eave width will change the drop off – see below 30° 22½° Smaller Drop-off (Higher Wall) Smaller eaves width Drop-off Same Roof Pitch Drop-off Larger eaves width Larger Drop-off (Lower Wall) Larger Roof Pitch Smaller eaves width SAME WALL HEIGHT Smaller Roof Pitch Larger eaves width Same Wall Height

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Exercises Calculate the drop off, assume 20mm for birdsmouth cut. 400 110, 40, 100 26° Tan 26° = opposite side 400 + 110 + 40 Tan 26° = opposite side 550 Re-arrange formula Opp side = tan 26° x 550 = 268mm Total drop off = 268 + 20 = 288 mm

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Exercises Calculate the drop off, assume 20mm for birdsmouth cut. 350 110, 50, 90 22° Tan 22° = opposite side 350 + 110 + 50 Tan 22° = opposite side 510 Re-arrange formula Opp side = tan 22° x 510 = 206 mm Total drop off = 206 + 20 = 226 mm

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Exercises Calculate the drop off, assume 20mm for birdsmouth cut. 300 110, 60, 100 36° Tan 36° = opposite side 300 + 110 + 60 Tan 36° = opposite side 470 Re-arrange formula Opp side = tan 36° x 470 = 341 mm Total drop off = 341 + 20 = 361 mm

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Exercises Calculate the drop off, assume 20mm for birdsmouth cut. 250 110, 40, 90 32° Tan 32° = opposite side 250+ 110 + 40 Tan 32° = opposite side 400 Re-arrange formula Opp side = tan 32° x 400 = 250 mm Total drop off = 250 + 20 = 270 mm

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