1. 11-12-13 Pythagorean Theorem, Special angles, and Trig Triangles Right Triangles Test Review

2. Pythagorean Theorem

3. Terminology We call the two sides that touch the right angle the LEGS. They are the a and b. The side opposite the right angle is the hypotenuse. It is the c. Label each part.

4. Terminology The hypotenuse is always the longest side but shorter than the two legs put together. Here is the 60 Second Review60 Second Review

5. Steps to Solve If you know the 2 legs – square the numbers – add- take square root.

6. Steps to Solve If you know the hypotenuse and one leg – square the numbers - subtract – take the square root.

7. Special Right Angles 45 – 45- 90

8. Special Right Angles If we represent the legs of an isosceles right triangle by 1, we can use the Pythagorean Theorem to establish pattern relationships between the lengths of the legs and the hypotenuse. These relationships will be stated as "short cut formulas" that will allow us to quickly arrive at answers regarding side lengths without applying trigonometric functions, or other means. There are two pattern formulas that apply ONLYto the 45º-45º-90º triangle.

9. Special Right Angles

11. Pythagorean Theorem, Special angles, and Trig Triangles What if I forget the pattern formulas? What should I do? Let's look at 3 solutions to this problem where you are asked to find x: Pattern Formula solution Pythagorean Theorem solution Trigonometric solution We are looking for the hypotenuse so we will use the pattern formula that will give the answer for the hypotenuse: Substituting the leg = 7, we arrive at the answer: A nice feature of the pattern formulas is that the answer is already in reduced form. Since a 45º-45º-90º, also called an isosceles right triangle, has two legs equal, we know that the other leg also has a length of 7 units. c 2 = a 2 + b 2 x 2 = 7 2 +7 2 x 2 = 49 + 49 x 2 = 98 Use either 45º angle as the reference angle. One possible solution is shown below: rounded

12. Special Right Angles

13. If you draw an altitude in an equilateral triangle, you will form two congruent 30º- 60º- 90º triangles. Starting with the sides of the equilateral triangle to be 2, the Pythagorean Theorem will allow us to establish pattern relationships between the sides of a 30º- 60º- 90º triangle. These relationships will be stated here as "short cut formulas" that will allow us to quickly arrive at answers regarding side lengths without applying trigonometric functions, or other means. There are three pattern relationships that we can establish thatapply ONLY to a 30º-60º-90º triangle.apply

14. Special Right Angles Labeling: H = hypotenuse LL = long leg (across from 60º) SL = short leg (across from 30º) Again 1 st step is to LABEL what you have!!

15. Short Cut Pattern Formulas short leg: You must remember that these formula patterns can be used ONLY in a 30º-60º-90º triangle. hypotenuse: H = 2SL Long leg:

16. Easy Practice x is the short leg Answer y is the long leg LL = SL Answer

17. Harder Practice 6 is the short leg and x is the hypotenuse (start with what you have given) H = 2 (SL) x = 2 (6) Answer y is the long leg LL = SL y = 6 Answer

18. Hardest Practice ( bonus?) 8 is the long leg and x is the hypotenuse (start with what you have given) Answer x =16/3 y is the short leg Answer

19. Right Δ Trigonometry

20. Label Hypotenuse 1 st. Then label side opposite of angle. The side left is your adjacent side. Right Δ Trigonometry MUST LABEL SIDES FROM ANGLE POINT OF VIEW MUST LABEL SIDES FROM ANGLE POINT OF VIEW

21. Right Δ Trigonometry

22. With both the TI 83 and 84 you need to make sure you are in degrees. With yellow TI 84, you need a ^^ after the angles to tell the calculator you are in degrees. But you do not need anything to solve for the inverse operations. Zoom equation takes care of it.

23. Right Δ Trigonometry There are actually 6 functions with trignonmetry. However, the cosecant, secant, and cotangent are just the reciprocal of the sine, cosine, and tangent. When we do not know the degree of the angle, we can calculate the fraction ratio, then find the inverse of the ratio with the 2 nd button on the calculator. This isn’t needed in zoom 400 and above. But in 300, divide the fraction, push 2 nd then your 9 sine, cosine, or tangent button, then 2 nd and (-) button for the answer of the fraction above.

24. Angle of Elevation The angle of elevation is always measured from the ground up. Think of it like an elevator that only goes up. It is always INSIDE the triangle. In the diagram at the left, x marks the angle of elevation of the top of the tree as seen from a point on the ground. You can think of the angle of elevation in relation to the movement of your eyes. You are looking straight ahead and you must raise (elevate) your eyes to see the top of the tree.

25. Angle of Depression The angle of depression is always OUTSIDE the triangle. It is never inside the triangle. In the diagram at the left, x marks the angle of depression of a boat at sea from the top of a lighthouse. diagram You can think of the angle of depression in relation to the movement of your eyes. You are standing at the top of the lighthouse and you are looking straight ahead. You must lower (depress) your eyes to see the boat in the water. As seen in the diagram above of angle of depression, the dark black horizontal line is parallel to side CA of triangle ABC. This forms what are called alternate interior angles which are equal in measure (so, x also equals the measure of <BAC). Simply stated, this means that: the angle of elevation = the angle of depression.

26. Let’s Practice

30. Things to Remember