Unit 4: Right Triangles Triangle Inequality Pythagorean Theorem and its Converse Trigonometry Inverse Trigonometry Solving Right Triangles

Lesson 4.1 Triangle Inequality Converse of the Pythagorean Theorem More Classifying Triangles

Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Practice the Triangle Inequality Can the following lengths represent the sides of a triangle? 5, 4, 3 Yes, add any two together and they are larger than the third side. 2) 5, 6, 7 5, 5, 10 No, 5+5 is equal to 10, not greater than 10.

Special Parts in a Right Triangle Right triangles have special names that go with it parts. For instance: The two sides that form the right angle are called the legs of the right triangle. The side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle. hypotenuse legs

Pythagorean Theorem c a b c2 = a2 + b2 c is always the hypotenuse a and b are the legs in any order c a b

Converse of the Pythagorean Theorem If c2 = a2 + b2 is true, then the triangle in question is a right triangle. You need to verify the three sides of the triangle given will make the Pythagorean Theorem true when plugged in. Remember the largest number given is always the hypotenuse Which is c in the Pythagorean Theorem

Acute Triangles from Pythagorean Theorem If c2 < a2 + b2, then the triangle is an acute triangle. So when you check if it is a right triangle and the answer for c2 is smaller than the answer for a2 + b2, then the triangle must be acute It essentially means the hypotenuse shrunk a little! And the only way to make it shrink is to make the right angle shrink as well! a b c

Obtuse Triangles from Pythagorean Theorem If c2 > a2 + b2, then the triangle is an obtuse triangle. So when you check if it is a right triangle and the answer for c2 is larger than the answer for a2 + b2, then the triangle must be obtuse It essentially means the hypotenuse grew a little! And the only way to make it grow is to make the right angle grow as well! a b c

Practice Determine if the following sides create triangle. If they do determine if it is a right, obtuse, or an acute triangle. A) 38, 77, 86 B) 10.5, 36.5, 37.5 c longest side Triangle Y/N Triangle Y/N c2 = a2 + b2 c2 = a2 + b2 862 = 382 + 772 37.52 = 10.52 + 36.52 7396 = 1444 + 5929 1406.25 = 110.25 + 1332.25 7396 = 7373 7396 > 7373 1406.25 < 1442.5 1406.25 = 1442.5 obtuse acute

Right Triangle?

Lesson 4.3 Trigonometric Ratios

Trigonometric Ratios A trigonometric ratio is a ratio of the lengths of any two sides in a right triangle. You must know: one angle in the triangle other than the right angle one side (any side) of the triangle. These help find any other side of the triangle.

Sine The sine is a ratio of c a b side opposite the known angle, and… the hypotenuse Abbreviated sin This is used to find one of those sides. Use your known angle as a reference point a b c θ side opposite θ b sin θ = = c hypotenuse

Cosine The cosine is a ratio of c a b side adjacent the known angle, and… the hypotenuse Abbreviated cos This is used to find one of those sides. Use your known angle as a reference point a b c θ side adjacent θ a cos θ = = c hypotenuse

Tangent The tangent is a ratio of c a b side opposite the known angle, and… side adjacent the known angle Abbreviated tan This is used to find one of those sides. Use your known angle as a reference point a b c θ side opposite θ b tan θ = = side adjacent θ a

SOHCAHTOA S o h C a T in This is a handy way of remembering which ratio involves which components. Remember to start at the known angle as the reference point. Also, each combination is a ratio So the sin is the opposite side divided by the hypotenuse pposite ypotenuse os djacent ypotenuse an pposite djacent

If you do not have a calculator with trig buttons, then turn to p845 in book for a table of all trig ratios up to 90o. Example 5 First determine which trig function you want to use by identifying the known parts and the variable side. Use that function on your calculator to find the decimal equivalent for the angle. Set that number equal to the ratio of side lengths and solve for the variable side using algebra. 37o 42o 4 x x 7 Get x out of denominator first by multiplying both sides by x. x 4 sin 42o = cos 37o = 7 x 7 (sin 42o) = x x (cos 37o) = 4 4 = 4 .7986 7 (.6691) = x = 4.683 x = = 5.008 cos 37o

Lesson 4.4 Inverse Trigonometric Ratios: Solving for missing angles in a right triangle.

Finding Angle Measures Inverse Trig Ratios Inverse trig ratios are used to find the measure of the angles of a triangle. The catch is…you must know two side lengths. Those sides determine which ratio to used based on the same ratios we had from before. Finding Side Lengths Finding Angle Measures sin sin-1 cos cos-1 tan tan-1 SOHCAHTOA

Example 8 You still base your ratio on what sides are you working with compared to the angle you want to find. Only now, your variable is θ. So once you find your ratio, you will then use the inverse function of your ratio from your calculator θ 4 7 θ 9 17 4 7 sin θ = 9 17 cos θ = 4 7 θ = sin-1 9 17 θ = cos-1 θ = sin-1 .5714 θ = cos-1 .5294 SOHCAHTOA θ = 34.8o θ = 58.0o