Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides

Transcription

1 Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = or 2 = 2-2 or 2 = 2-2 The re of the squre on the hypotenuse is equl to the sum of the squres on the other two sides Pge of 8

2 Pythgors theorem nd trigonometry Pythgors Theorem Find the length of the missing sides 6m You dd to find the hypotenuse You sutrt to find shorter side 0m 8m 5m 2 = = = = 00 = 00 = 0m 2 = = = = 25 = 25 =.2m (dp) 2 = = = 2-2. Lel the tringle 2. Deide whether to + or - + for hypotenuse - for shorter side 3. Squre the numers 4. Squre root your nswer 5. Put on the orret units If you don t hve lultor, leve your nswer s squre root Pge 2 of 8

3 Pythgors theorem nd trigonometry Trigonometry The study of tringles is lled Trigonometry Opposite The side opposite the ngle θ is the ngle you re going to use ypotenuse djent djent The side next to the ngle θ The ypotenuse is the longest side Opposite ypotenuse The ypotenuse is lwys opposite the right-ngle S O T O S = O Opposite S in θ = ypotenuse Sin O = djent θ os θ = Tn θ = ypotenuse os nuse T = O Opposite djent Tn O Pge 3 of 8

4 Pythgors theorem nd trigonometry Trigonometry Period 360 o Grph of y = sinx.5 y Osilltes etween nd x Period 360 o Grph of y = osx.5 y Osilltes etween nd x Period 80 o.5 Grph of y = tnx 0 y symptotes t 270 o, -90 o, 90 o, 270 o 5 x Pge 4 of 8

5 Trigonometry Pythgors theorem nd trigonometry Exmple Find the length of side p. p O Exmple 2 2m 37 o Find the size of ngle. 3m O 4.5m S O T O Sin = O Sin 37 o = p 2 2 x Sin 37 o = p p = 7.2m (dp) S O T O Tn = O Tn = = Tn = 56.3 o (dp) Method. Lel the sides, O,,. 2. Write down S O T O 3. Deide whih two sides you re using, then selet Sin, os or Tn 4. Write the trig. funtion with the informtion for your tringle. 5. lulte. 6. hek your nswer is sensile. Exmple 3 ldder of 4 metres in length is resting ginst wll. The ldder mkes n ngle of 60 o with the ground. () Find how high up the wll the ldder rehes, h. () Find the distne from the wll to the ottom of the ldder, d. () S O T O () S O T O 4m 60 o h Sin = O Sin 60 o = h 4 4 x Sin 60 o = h h = 3.46m (2dp) os = os 60 o = d 4 4 x os 60 o = d d = 2m One you know h you ould use Pythgors rule to lulte d d Pge 5 of 8

6 Trigonometry Pythgors theorem nd trigonometry S O T O 3 30 o Lern: Sin 30 o = 2 60 o 2 Sin 30 o = 2 = OPP YP OPP = YP = 2 Using Pythgors DJ must = 3 Sin 30 o = 2 os 60 o = 2 Sin 60 o = 3 2 os 30 o = 3 2 Tn 60 o = 3 Tn 30 o = 3 = 3 Lern: Tn 45 o = = Tn 45 o = = = OPP DJ OPP = DJ = Using Pythgors YP Sin 45 o = 2 must = 2 You must lel the tringle O,, for the ngle you re using. hnging the ngle will men repositioning the lels 2 45 o os 45 o = 2 45 o Pge 6 of 8

7 Pythgors theorem nd trigonometry The osine Rule It is used with ny non right-ngled tringle To find side 2 = os or os = NGLES re mrked with PITLS nd the opposite side with the sme lower se letter To find n ngle Use this rule when given two sides nd the ngle etween them or ll three sides nd no ngles 6m 3m 0m 73 o 7m 8m INV os Exmple Find the length of side Use 2 = os 2 = x 6 x 7 x os73 o 2 = Pge 7 of 8 = = 7.8m (dp) Exmple 2 Find ngle Use os = 2 os = x 3 x 0 os = = os - ( ) = 38.0 o (dp) = Use full lultor vlues to the end

8 Pythgors theorem nd trigonometry It is used with ny non right-ngled tringle The Sine Rule NGLES re mrked with PITLS nd the opposite side with the sme lower se letter To find side put the sides on top = = sin sin sin or sin sin sin = = To find n ngle put the ngles on top 68 o 8m 7m Use this rule when you n prtner one ngle nd side nd need to find the ngle or side from nother pir Prtners re opposite eh other 40 o 9.5m Exmple Find ngle Exmple 2 Find side 54 o INV sin sin Use = sin sin = sin68 o 7 8 sin = 7 x sin68 o 8 sin = = sin - ( ) = 54.2 o (dp) Use sin = sin = 9.5 sin54 o sin40 o 9.5 x sin54 sin40 o = 2.0m (dp) Use full lultor vlues to the end Pge 8 of 8