1. When 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) A C B a b

2. If you know two of the sides, you can use the Pythagorean theorem to find the other side A = 3 C B = 4 a b

3. And 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 90º For example: if a = 30º b = 90º – 30º b = 60º A C B a b

4. But 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(A) k(C) k(B) a b A C B a b

5. How does that help us? 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 shape of triangle! A = 1 C = 2 B 30º 60 º

6. But there are three pairs of sides possible! 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

7. So what are the formulas? SOHCAHTOA Sin is Opposite over HypotenuseCos is Adjacent over Hypotenuse Tan is Opposite over Adjacent

8. Some terminology: 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

9. A picture always helps… looking at the triangle in terms of angle b A C B b adj hyp opp b C is always the hypotenuse A is the adjacent (near the angle) B is the opposite (across from the angle) Longest Near Across

10. But if we switch angles… looking at the triangle in terms of angle a A C B a opp hyp adj a C is always the hypotenuse A is the opposite (across from the angle) B is the adjacent (near the angle) Longest Across Near

11. Lets try an example Suppose we want to find angle a what is side A? the opposite what is side B? the adjacent with opposite and adjacent we use the… tan formula A = 3 C B = 4 a b

12. Lets solve it A = 3 C B = 4 a b

13. Another tangent example… we want to find angle b B is the opposite A is the adjacent so we use tan A = 3 C B = 4 a b

14. Calculating a side if you know the angle you know a side (adj) and an angle (25 ° ) we want to know the opposite side A C B = 6 25° b

15. Another tangent example If you know a side and an angle, you can find the other side. C A = 6 25° b B

16. An application 65° 10m 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

17. Why do we need the sin & cos? 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! C = 10 A 25° b B

18. Lets do sin first we want to find angle a since we have opp and hyp we use sin C = 10 a b B A = 5

19. And one more sin example find the length of side A We have the angle and the hyp, and we need the opp C = 20 25° b B A

20. And finally cos We use cos when we need to work with the hyp and adj so lets find angle b C = 10 a b B A = 4

21. Here 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 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?

22. How do we know which formula to use??? Well, what are we working with? We have an angle We have hyp We need opp With these things we will use the sin formula C = 5 65° B

23. So lets calculate so Spike will have fallen 4.53m C = 5 65° B

24. One last example… 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?

25. First draw a triangle What parts do we have? We have an angle We have the Adjacent We need the opposite Since we are working with the adj and opp, we will use the tan formula 2m 88° B

26. So lets calculate Lucretia’s walkman fell 57.27m 2m 88° B

27. What are the steps for doing one of these questions? 1. Make a diagram if needed 2. Determine which angle you are working with 3. Label the sides you are working with 4. Decide which formula fits the sides 5. Substitute the values into the formula 6. Solve the equation for the unknown value 7. Does the answer make sense?

28. Two Triangle Problems 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.

29. Draw a diagram You 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 45m 60° 40 ° 45m

30. Break the problem into two triangles. The first triangle: The second triangle note that they share a side 45m long a and b are heights! 60° 45m 40 ° b a

31. The First Triangle We are dealing with an angle, the opposite and the adjacent this gives us Tan 60° 45m a

32. The second triangle We are dealing with an angle, the opposite and the adjacent this gives us Tan 45m 40 ° b

33. What does it mean? Look at the diagram now: the short building is 37.76m tall the tall building is 77.94m plus 37.76m tall, which equals 115.70m tall 60° 40 ° 45m 77.94m 37.76m