G E O M E T R Y CHAPTER 9 RIGHT TRIANGLES AND TRIGONOMETRY. Notes & Study Guide


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1 G E O M E T R Y CHAPTER 9 RIGHT TRIANGLES AND TRIGONOMETRY Notes & Study Guide
2 2 TABLE OF CONTENTS SIMILAR RIGHT TRIANGLES... 3 THE PYTHAGOREAN THEOREM... 4 SPECIAL RIGHT TRIANGLES... 5 TRIGONOMETRIC RATIOS... 7 SOLVING RIGHT TRIANGLES HOMEWORK EXAMPLES... 13
3 SIMILAR RIGHT TRIANGLES 3 One way that two triangles were similar was by the AA Postulate (2 angles from one congruent to 2 angles from another). From Chapter 8 For right triangles, it is possible to create three similar triangles (all proven by AA) by just adding one altitude. This is Theorem 9.1 Since the triangles are all similar, all angles are equal and all sides are proportional. To find any missing info, work the same way as Chapter 8. RECOMMENDED TECHNIQUE Separate the three triangles (makes identification easier) Create your proportion (be consistent with pieces you use) Use cross products to solve as usual. EXAMPLE Separate the triangles for easy identification Set up your proportion and do cross products
4 4 THE PYTHAGOREAN THEOREM In all right triangles, there is a pattern involving the lengths of the sides. The pattern is represented by one of the most recognized formulas in mathematics. PYTHAGOREAN THEOREM In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. EXAMPLES a 2 + b 2 = c 2 (c is the hypotenuse) PYTHAGOREAN TRIPLES Any three integers (no decimals) that make Pythagorean s Theorem true is called a Pythagorean Triple. Triples are shortcuts...if you recognize that one is there, you don t have to use the formula. MOST COMMON TRIPLES: 3, 4 and 5 5, 12 and 13 7, 24 and 25 CLASSIFYING TRIANGLES USING THE THEOREM Pythagoreans Theorem can be used to determine what type (acute/right/obtuse) of triangle that you have. You do it by comparing the c 2 part to the a 2 + b 2 part. c 2 < a 2 + b 2 c 2 = a 2 + b 2 c 2 > a 2 + b 2 acute right obtuse Substitute the value of the sides into the formula and see which one occurs.
5 SPECIAL RIGHT TRIANGLES 5 All right triangles fit the Pythagorean Theorem pattern (a 2 + b 2 = c 2 ). But there are two specific right triangles, called SPECIAL RIGHT TRIANGLES, that have another pattern in themselves. They are identified by their angle sizes Pythagorean s Theorem requires you know 2 sides to use it. Special Right Triangles only require one that s the shortcut TRIANGLE Short leg the side opposite the 30 angle Long leg the side opposite the 60 angle Hypotenuse the side opposite the 90 angle Hypotenuse = 2 short leg Long leg = 3 short leg The pattern is based on the short leg! TRIANGLE Legs the sides opposite the 45 angles Hypotenuse the side opposite the 90 angle Hypotenuse = 2 Leg Legs are equal
6 6 RIGHT TRIANGLE EXAMPLES Find the values of the missing sides of each right triangle. Use Pythagorean s Theorem to classify each triangle. Find the unknown side length in each right triangle. Round off any answers to 2 decimal places.
7 TRIGONOMETRIC RATIOS 7 All right triangles work with Pythagorean s Theorem, but you need 2 sides to use it. If you don t have 2 sides, special right triangles are used, but they are very specific. If you still don t have 2 sides and no special right triangle, that s where trig comes in. A trigonometric (trig) ratio is a ratio (fraction) that compares any two sides from a right triangle. For every different pair of sides used, you get a different trig ratio. The sides are named by their position to the specific angle being referenced. EXAMPLE: Angle A is the reference angle. Opposite directly across from the angle Adjacent next to the angle Hypotenuse across from right angle DEFINING THE TRIG RATIOS There are six different ways to create a ratio using any two sides from the right triangle. Each way is a trig ratio. The three primary ratios we will use are EXAMPLE sin A = cos A = tan A = sin B = cos B = tan B = Trig ratios are usually written as decimals to four decimal places!
8 8 TRIGONOMETRIC RATIOS HOW TRIG RATIOS RELATE TO ANGLES It is important to know that all trig ratios are unique. This means that every individual trig ratio corresponds to just one angle size. So when you find a trig ratio, that value can only exist for one specific size of angle. If the ratio changes, then the angle size does too. EXAMPLE: Say we find that sin A = , that ratio means A = 40 Say we find that cos B = , that ratio means B = 63.3 Say we find that tan C = , that ratio means C = 28 How do we know? Two ways (1) we use a scientific calculator (2) we use a TABLE OF TRIG VALUES that has all the ratios matched up to their angle sizes. It s on page 845 of your textbook. EXAMPLE 1 What is the sin A? According to the table that ratio means A = What is the size of angle B? (use any trig ratio you choose) EXAMPLE 2 Find the sizes of angles R and S. R = S =
9 SOLVING RIGHT TRIANGLES 9 Every right triangle has the follow pieces 3 angles (1 right, 2 acute) 3 sides (1 hypotenuse and 2 legs) Finding out the measure of all 3 angles and the lengths of all 3 sides is what is referred to as SOLVING A RIGHT TRIANGLE. HOW TO SOLVE A RIGHT TRIANGLE To solve a right triangle, you will make use of two primary techniques (1) Pythagorean s Theorem (used to find sides) (2) Trig Ratios (used to find angles) In most cases, you will need to use both methods (one time each) to completely solve a right triangle. EXAMPLE 1 Solve the right triangle shown. g = h = angle G = EXAMPLE 2 Solve the right triangle shown. c = angle A = angle B =
10 10 HOMEWORK EXAMPLES
11 HOMEWORK EXAMPLES 11
12 12 HOMEWORK EXAMPLES
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