2When you have a right triangle there are 5 things you can know about it.. the lengths of the sides (A, B, and C)the measures of the acute angles (a and b)(The third angle is always 90 degrees)bCAaB
3If you know two of the sides, you can use the Pythagorean theorem to find the other side bCA = 3aB = 4
4And if you know either angle, a or b, you can subtract it from 90 to get the other one: a + b = 90 This works because there are 180º in a triangle and we are already using up 90ºFor example:if a = 30ºb = 90º – 30ºb = 60ºbCAaB
5But what if you want to know the angles? Well, here is the central insight of trigonometry:If you multiply all the sides of a right triangle by the same number (k), you get a triangle that is a different size, but which has the same angles:k(C)bCbAk(A)aaBk(B)
6How does that help us?Take a triangle where angle b is 60º and angle a is 30ºIf side B is 1unit long, then side C must be 2 units long, so that we know that for a triangle of this shape the ratio of side B to C is 1:2There are ratios for everyshape of triangle!C = 260 ºA = 130ºB
7But there are three pairs of sides possible! Yes, so there are three sets of ratios for any triangleThey are mysteriously named:sin…short for sinecos…short for cosinetan…short or tangentand the ratios are already calculated, you just need to use them
8So what are the formulas? Tan is Opposite over AdjacentCos is Adjacent over HypotenuseSin is Opposite over HypotenuseSOHCAHTOAYou can use this word if youneed to memorize the formulas!
9Some terminology:Before we can use the ratios we need to get a few terms straightThe hypotenuse (hyp) is the longest side of the triangle – it never changesThe opposite (opp) is the side directly across from the angle you are consideringThe adjacent (adj) is the side right beside the angle you are considering
10A picture always helps… looking at the triangle in terms of angle bbA is the adjacent (near the angle)CAB is the opposite (across from the angle)BbNearC is always the hypotenusehypLongestadjoppAcross
11But if we switch angles… looking at the triangle in terms of angle aA is the opposite (across from the angle)CAaB is the adjacent (near the angle)BAcrossC is always the hypotenusehypLongestoppaadjNear
12Lets try an example Suppose we want to find angle a what is side A? the oppositewhat is side B?the adjacentwith opposite and adjacent we use the…tan formulabCA = 3aB = 4
14Where did the numbers for the ratio come from? Each shape of triangle has three ratiosThese ratios are stored your scientific calculatorIn the last question, tanθ = 0.75On your calculator try 2nd, Tan 0.75 = °
15Another tangent example… we want to find angle bB is the oppositeA is the adjacentso we use tanbCA = 3aB = 4
16Calculating a side if you know the angle you know a side (adj) and an angle (25°)we want to know the opposite sidebCA25°B = 6
17Another tangent example If you know a side and an angle, you can find the other side.bCA = 625°B
18An applicationYou look up at an angle of 65° at the top of a tree that is 10m awaythe distance to the tree is the adjacent sidethe height of the tree is the opposite side65°10m
19Why do we need the sin & cos? We use sin and cos when we need to work with the hypotenuseif you noticed, the tan formula does not have the hypotenuse in it.so we need different formulas to do this worksin and cos are the ones!bC = 10A25°B
20Lets do sin first we want to find angle a since we have opp and hyp we use sinbC = 10A = 5aB
21And one more sin example find the length of side AWe have the angle and the hyp, and we need the oppbC = 20A25°B
22And finally cos We use cos when we need to work with the hyp and adj so lets find angle bbC = 10A = 4aB
23Here is an example Spike wants to ride down a steel beam The beam is 5m long and is leaning against a tree at an angle of 65° to the groundHis friends want to find out how high up in the air he is when he starts so they can put add it to the doctors report at the hospitalHow high up is he?
24How do we know which formula to use??? Well, what are we working with?We have an angleWe have hypWe need oppWith these things we will use the sin formulaC = 5B65°
25So lets calculateC = 5Bso Spike will have fallen 4.53m65°
26One last example…Lucretia drops her walkman off the Leaning Tower of Pisa when she visits ItalyIt falls to the ground 2 meters from the base of the towerIf the tower is at an angle of 88° to the ground, how far did it fall?
27First draw a triangle What parts do we have? We have an angle We have the AdjacentWe need the oppositeSince we are working with the adj and opp, we will use the tan formulaB88°2m
29What are the steps for doing one of these questions? Make a diagram if neededDetermine which angle you are working withLabel the sides you are working withDecide which formula fits the sidesSubstitute the values into the formulaSolve the equation for the unknown valueDoes the answer make sense?
30Two Triangle ProblemsAlthough there are two triangles, you only need to solve one at a timeThe big thing is to analyze the system to understand what you are being givenConsider the following problem:You are standing on the roof of one building looking at another building, and need to find the height of both buildings.
31Draw a diagramYou can measure the angle 40° down to the base of other building and up 60° to the top as well. You know the distance between the two buildings is 45m60°40°45m
32Break the problem into two triangles. The first triangle:The second trianglenote that they share a side 45m longa and b are heights!a60°45m40°b
33The First TriangleWe are dealing with an angle, the opposite and the adjacentthis gives us Tana60°45m
34The second triangleWe are dealing with an angle, the opposite and the adjacentthis gives us Tan45m40°b
35What does it mean? Look at the diagram now: the short building is 37.76m tallthe tall building is 77.94m plus 37.76m tall, which equals m tall77.94m60°40°37.76m45m