1. Vectors Right Triangle Trigonometry

2. 9-1 The Tangent Ratio  The ratio of the length to the opposite leg and the adjacent leg is the Tangent of angle A A C B Angle A Leg opposite angle A Leg adjacent to angle A

3. Writing the Tangent  The tangent of angle A is written as  tanA =

4. Identifying Tangents tanA = tanB = A B C 12 5 13

5. Tangent Inverse  The Tangent Inverse allows you to find the angle given the opposite and adjacent sides from this angle.  X=Tan -1 (2/5) x 2 5

6. 9-2 Sine and Cosine Ratios Leg opposite angle A Leg adjacent to angle A Hypotenuse Angle A

7. Sine and Cosine 15 8 17 A B C

8. Sin -1 and Cos -1  Angle A = sin -1 (8/17)  Angle B = cos -1 (15/17) A C B 15 17 8

9. Keeping It Together  Use the following acronym to help you remember the ratios  SOHCAHTOA  Sine is Opposite over Hypotenuse  Cosine is Adjacent over Hypotenuse  Tangent is Opposite over Adjacent

10. 9-3 Angles of Elevation & Depression  Angle of Elevation- measured from the horizon up  Angle of Depression- measured from the horizon down

11. Angle of elevation x The angle of elevation is the angle formed by the line of sight and the horizontal

12. Angle of depression x The angle of depression is the angle formed by the line of sight and the horizontal

13. Combining the two x x elevation depression It’s alternate interior angles all over again!

14. B A 21 h m The angle of elevation of building A to building B is 25 0. The distance between the buildings is 21 meters. Calculate how much taller Building B is than building A. Step 1: Draw a right angled triangle with the given information. Step 3: Set up the trig equation. Angle of elevation Step 4: Solve the trig equation. 25 0 Step 2: Take care with placement of the angle of elevation

15. Step 1: Draw a right angled triangle with the given information. Step 3: Decide which trig ratio to use. 60 m 80 m   Step 4: Use calculator to find the value of the unknown. A boat is 60 meters out to sea. Madge is standing on a cliff 80 meters high. What is the angle of depression from the top of the cliff to the boat? Step 2: Use your knowledge of alternate angles to place  inside the triangle. Angle of depression

16. 9-4 Vectors  Vector - a quantity with magnitude (the size or length) and direction, it is represented by an arrow  Initial Point- is where the vector starts, i.e., the tail of the arrow  Terminal Point- is where the arrow stops, i.e., the point of the arrow

17. Vectors  The magnitude corresponds to the distance from the initial point to the terminal point. The symbol for the magnitude of a vector is.  The symbol for a vector is an arrow over a lower case letter, or capital letters of the initial and terminal points  The distance corresponds to the direction in which the arrow points

18. Describing Vectors  An ordered pair in a coordinate plane can also be used for a vector.  The magnitude is the cosine and the direction is the sine. The ordered pair is written this way,, to indicate a vectors distance from the origin.  A vector with the initial point at the origin is said to be in Standard Position.

19. Describing Vectors in the Coordinate Plane  With a vector in Standard Position, the coordinates of the terminal point describes the vector.  The magnitude is the hypotenuse of a right triangle. The cosine of the direction angle is the x coordinate and the sine is the y coordinate  See Example 1 on Pg. 490

20. Describing a Vector Direction  Vector direction commonly uses compass directions to describe a vector.  The direction is given as a number of degrees east, west, north or south of another compass direction, such as 25 0 east of north  See Example 2 Pg. 491

21. Vector Addition  A vector sum is called the RESULTANT.  Adding vectors gives the result of vectors that occur in a sequence (See the top of pg. 492) or that act at the same time (See Examples 4 & 5 pgs. 492, 493)

22. 9-5 Trig Ratios and Area  Parts of Regular Polygons Center- a point equidistant from the vertices Radius- a segment from the center to a vertex Apothem- a segment from the center perpendicular to a side Central Angle- angle formed by two radii

23. Finding Area in a Regular Polygon  Formula for Area A=(apothem X perimeter) divided by 2  Use the trig ratio, and the central angle to find the apothem or a side for the perimeter.  See Examples 1 & 2 pgs. 498-499

24. Area of a Triangle Given SAS  Theorem 9-1 The area of a triangle is one half the product of the lengths of the sides and the sine of the included angle. Where b and c are sides and A is the angle between them. See the bottom of pg 499 and Example 3 pg. 500