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) b C A a B
A C2 B2B C2 A2C A2 B2if A 3, B 4C A2 B2 b CC 32 42 A=3 aC 25 5 B=4
This works because there are 180º in a triangle and we are already using up 90º For example: if a = 30º b = 90º – 30º b b = 60º C A a B
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) b C b A k(A) a a B k(B)
Take a triangle where angle b is 60º and angle a is 30º If 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 1:2 There are ratios for every C=2 60 ºshape of triangle! A=1 30º B
Yes, so there are three sets of ratios for any triangle They are mysteriously named: sin…short for sine cos…short for cosine tan…short or tangent and the ratios are already calculated, you just need to use them
oppsin Tan is Adjacent over Hypotenuse Cos is Opposite over Adjacent Sin is Opposite over Hypotenuse hyp adj SOHCAHTOAcos hyp opptan adj
Before we can use the ratios we need to get a few terms straight The hypotenuse (hyp) is the longest side of the triangle – it never changes The opposite (opp) is the side directly across from the angle you are considering The adjacent (adj) is the side right beside the angle you are considering
looking at the triangle in terms of angle b A is the adjacent b C (near the angle) A B is the opposite B (across from the angle) b Near C is always the Longest hyp hypotenuse adj opp Across
looking at the triangle in terms of angle a A is the opposite (across C from the angle) A a B is the adjacent (near B the angle) hyp Across C is always the Longest hypotenuse opp a adj Near
Suppose we want to find angle a opp what is side A? tan the opposite adj what is side B? the adjacent with opposite and adjacent we use C b the… A=3 tan formula a B=4
opptan adj 3tan a 0.75 4check our calculator s ba 36.87º C A=3 a B=4
Each shape of triangle has three ratios These ratios are stored your scientific calculator In the last question, tanθ = 0.75 On your calculator try 2nd, Tan 0.75 = 36.87
we want to find angle b opp B is the opposite tan A is the adjacent adj so we use tan 4 tan b 3 b C tan b 1.33 A=3 a b 53.13 B=4
you know a side (adj) and an angle (25 ) we want to know the opposite side opp tan adj Atan 25 6A tan 25 6 bA 0.47 6 C AA 2.80 25 B=6
If you know a side and an angle, you can find the other side. 6 opp tan 25 tan B adj 6 B tan 25 b 6 C B A=6 0.47 25 B 12.87 B
You look up at an angle of 65° at the top of a tree that is 10m away the distance to the tree is the adjacent side the height of the tree is the opposite side opp tan 65 10 opp 10 tan 65 65 opp 10 2.14 10m opp 21.4
We use sin and cos when we need to work with the hypotenuse if you noticed, the tan formula does not have the hypotenuse in it. so we need different formulas to do this work sin and cos are the ones!10 C= b A 25 B
we want to find angle a since we have opp and hyp opp we use sin sin hyp 5sin a 10 C = 10 bsin a 0 .5 A=5a 30 a B
find the length of side A opp We have the angle sin hyp and the hyp, and we need the opp Asin 25 20 A sin 25 20 C = 20 b A 0.42 20 A A 8.45 25 B
We use cos when we need to work with the hyp and adj adj so lets find angle b cos hyp 4cos b b 10 C = 10 A=4cos b 0.4 ab 66.42 B a 90 - 66.42 a 23.58
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 ground His 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 hospital How high up is he?
Well, what are we working with? We have an angle We have hyp C=5 We need opp B With these things we will use the sin formula 65
oppsin 65 hyp oppsin 65 5 C=5opp sin 65 5 Bopp 0.91 5opp 4.53 so Spike will have fallen 65 4.53m
Lucretia drops her walkman off the Leaning Tower of Pisa when she visits Italy It falls to the ground 2 meters from the base of the tower If the tower is at an angle of 88 to the ground, how far did it fall?
What parts do we have? We have an angle We have the Adjacent We need the opposite Since we are working with B the adj and opp, we will use the tan formula 88 2m
1. Make a diagram if needed2. Determine which angle you are working with3. Label the sides you are working with4. Decide which formula fits the sides5. Substitute the values into the formula6. Solve the equation for the unknown value7. Does the answer make sense?
Although there are two triangles, you only need to solve one at a time The big thing is to analyze the system to understand what you are being given Consider 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.
You can measure the angle 40° down to the base of other building and up 60° to the top as well. 60 You know the 40 distance between the two buildings is 45m 45m
The first triangle: a 60 The second 45m triangle 40 b notethat they share a side 45m long
We are dealing with an angle, the opposite and the adjacent this gives us Tan a tan 60 45 a tan 60 45 a a 1.73 45 a 77.94m 60 45m
We are dealing with an angle, the opposite and the adjacent this gives us Tan b tan 40 45m 45 40 b tan 40 45 b b 0.84 45 b 37.76m
Look at the diagram now: the short building is 37.76m tall 77.94m the tall building is 60 77.94m plus 37.76m 40 tall, which equals 115.70m tall 37.76m 45m