Right Triangles and Trigonometry Similar Triangles are characterized by congruent corresponding angles and proportionate corresponding sides. There are several distinct characteristics of Right Triangles that we will look at in Chapter 9 that make them very useful in designing structures and analyzing the world around us. Theorem 9.1 If the altitude is drawn from the right angle to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle, and to each other /\ CBD ~ /\ ABC, /\ ACD ~ /\ ABC, and /\CBD ~ /\ACD C A D B

Finding the Height of a Roof Y 5.5 h 3.1 Z W X |-------------------------6.3---------------------------| The roof has a cross section that forms a right triangle. How can we use that information to determine the actual height of the roof? 1. Create a series of “similar” right triangles Z Y ~ ~ X W Y W X Y 6.3 5.5 5.5 3.1 h h 3.1 2. Then use the proportional component of similarity to determine the value of h YW – XY ZY XZ _h_ -- 3.1 5.5 6.3 6.3h = 5.5(3.1) = 2.7

Using Geometric Mean to Solve Problems Theorem 9.2 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the GEOMETRIC MEAN of the lengths of the two segments. C A D B BD – CD CD AD Theorem 6.3 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. AB – CB CB DB AB – AC AC AD

Using Geometric Mean 6 – x x 3 18 = x2 X = \/18 X = 3\/2 2 x y 5 6 3 6 3 6 – x x 3 18 = x2 X = \/18 X = 3\/2 5 + 2 – y y 2 7 – y Y 2 Y2 = 14 Y = \/14

SOHCAHTOA Trigonometric Ratios Triangle ABC is a right triangle. The sine (sin), cosine (cos) and tangent (tan) are defined as follows: sin A = side opposite <A = a hypotenuse c cos A = side adjacent <A = b hypotenuse c tan A = side opposite <A = a side adjacent <A b B c a A b C Hypotenuse side opposite <A side adjacent <A SOHCAHTOA

Finding Trig Ratios Based upon the SSS Similarity Theorem the two triangles are similar. What impact do you think this will have on the Trigonometric ratios? 17 8 4 8.5 15 7.5 B C A C A Large Triangle Small Triangle 8/17 = 0.4706 4/8.5 = 0.4706 15/17 = 0.8824 15/17 = 0.8824 8/15 = 0.5333 4/7.5 = 0.533 sin A = side opposite <A = a hypotenuse c cos A = side adjacent <A = b hypotenuse c tan A = side opposite <A = a side adjacent <A b

Trig Ratios for 45o and 30o Angles In a 45 – 45 – 90 Triangle, the Sin, Cos, and Tan will always fall into a standard proportion to one another based upon the standard relationship between the sides (9.4). 1 \/ 2 45o 1 sin 45o = 1/ \/2 = \/2 / 2 = 0.7071 cos 45o = 1/ \/2 = \/2 / 2 = 0.7071 tan 450 = 1/1 = 1 In a 30 - 60 – 90 Triangle, the Sin, Cos, and Tan will always fall into a standard proportion to one another based upon the standard relationship between the sides (9.4). 1 2 30o \/3 sin A = side opposite <A = a hypotenuse c cos A = side adjacent <A = b hypotenuse c tan A = side opposite <A = a side adjacent <A b sin 30o = 1 / 2 = 0.5 cos 30o = \/3 / 2 = 0.8660 tan 30o = 1 / \/3 = \/3 / 3 = 0.574

Solving for a Right Triangle Solving for a Right Triangle means determining the measures of every side and every angle - We can do this if we know: * The measure of 2 Sides, or * The measure of 1 Side and 1 Angle Solving knowing 1 Side & 1 Angle If we know the measure of one angle and one side, we can calculate the length of the missing sides. Given we know the measure of the acute angle, we can determine the missing angle by subtracting it from 90 Triangle XYZ has one acute angle that measures 25o and a hypotenuse that measures 13 Taking the Sin of X, we determine the opposite side to have a measure of 5.5 Taking the Cos of X we determine the adjacent side to be 11.8. Therefore the triangle has side measures of 5.5, 11.8, and 13; and angle measures of 25, 65, and 90 Solving knowing 2 sides: If we know the measure of two sides, we can calculate the Sin, Cos, or Tan (depending on givens) Once we know that measure, we can take the inverse of the measure to determine the measure of the angle Triangle ABC has leg measures of 2 and 3 Taking 2/3 we determine that the TAN of angle A = .6 The inverse of the Tan of .6 = 33.7 Therefore the measure of angle A = 33.7 Therefore Triangle ABC has side measures of 2, 3, and Root 13; and angle measures of 90, 33.7, and 56.3 B 2 C 3 A Y h 13 25o Z g X

Trig Application – Glide Angles and Glide Ratios Aerospace Design and landing takes into account two concepts based upon Basic Trig Ratios Glide Angle: The angle of approach taken by an aircraft as it enters its landing pattern Using this angle, pilots and ATC can safely guide a plan on a steady descent to it’s landing Glide Ratio: Mathematical calculation based on the design of the aircraft that under specific conditions (primarily determined by speed and altitude) Determines the distance a plane will travel under “glide” conditions When an aircraft (Space Shuttle) is at an altitude of 15.7 Miles it is 59 miles away from the runway. What is the glide angle of the approach? TAN x = 15.7 / 59 = 14.9o When the shuttle is 5 miles away, it has increased it’s glide angle to 19o. What is it’s altitude? TAN 19o = h/5 = 1.7 Miles