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Vectors Right Triangle Trigonometry 9-1 The Tangent Ratio The ratio of the length to the opposite leg and the adjacent leg is the B Tangent of angle A Leg opposite angle A Angle A A C Leg adjacent to angle A Writing the Tangent The tangent of angle A is written as tanA = opposite adjacent Identifying Tangents 5 tanA = 12 B 12 tanB = 5 13 A 12 5 C 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 21.8 0 2 x 5 9-2 Sine and Cosine Ratios opposite sin A hypotenuse adjacent cos A hypotenuse Leg opposite angle A Angle A Leg adjacent to angle A Sine and Cosine 8 sin A 17 B 17 15 cos A 17 8 A C 15 Sin-1 and Cos-1 Angle A = sin-1(8/17) Angle _ A 28.07 Angle B = 0 B cos-1(15/17) Angle _ A 28.07 17 0 A 15 8 C 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 9-3 Angles of Elevation & Depression Angle of Elevation- measured from the horizon up Angle of Depression- measured from the horizon down Angle of elevation The angle of elevation is the angle formed by the line of sight and the horizontal x Angle of depression x The angle of depression is the angle formed by the line of sight and the horizontal Combining the two x depression It’s alternate interior angles all over again! elevation x The angle of elevation of building A to building B is 250. 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 2: Take care with placement of the angle of elevation Step 3: Set up the trig equation. Step 4: Solve the trig equation. Angle of elevation B hm A 250 21 tan 25 h 21 h 21 tan 25 h 9.8 m (1 dec. pl ) 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 1: Draw a right angled triangle with the given information. Step 2: Use your knowledge of 80 m alternate angles to place inside the triangle. Step 3: Decide which trig ratio to use. Step 4: Use calculator to find the value of the unknown. Angle of depression 60 m 80 60 1 80 tan 60 tan 53.1o 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 Vectors The magnitude corresponds to the distance from the initial point to the terminal point. The symbol for the magnitude of a vector is V . The symbol for a vector is an arrow over a lower case letter, a or capital letters of the initial and terminal points The distance corresponds to the direction in which the arrow points 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, x, y , to indicate a vectors distance from the origin. A vector with the initial point at the origin is said to be in Standard Position. 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 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 250 east of north See Example 2 Pg. 491 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) 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 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 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. bc(sin A) A 2 Where b and c are sides and A is the angle between them. See the bottom of pg 499 and Example 3 pg. 500