Topic 8 Goals and common core standards Ms. Helgeson Solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle. Use a geometric mean to solve problems. Prove the Pythagorean Thm Use the Converse of the Pythagorean Thm to solve problems Use side lengths to classify triangles by their angle measures.

Find the side lengths of special right triangles. Find the sine, the cosine, and the tangent of an acute angle. Use trig ratios to solve real-life problems Solve a right triangle Use right triangles to solve real-life problems CC.9-12.G.SRT.4 CC.9-12.G.SRT.6

CC.9-12.G.SRT.8 CC.9-12.G.SRT.10(+) CC.9-12.G.SRT.11(+)

Right Triangles and Trigonometry Chapter 9 Right Triangles and Trigonometry

Radical Review Pages 522 - 523

8.1 Right Triangles and The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. C = a + b 2 2 2

Pythagorean triple A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation C = a + b Ex: The integers 3, 4, and 5; 5, 12, 13; 8, 15, 17; 7, 24, 25 form the family of Pythagorean triples. 2 2 2

Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. Is this a right triangle with sides: 33, 65, 56? Is this a right triangle with sides: 8√6, 24, and 8√15

Use similarity to prove the pythagorean theorem Use similarity to prove the pythagorean theorem. Given: ∆XYZ is a right triangle Prove: a² + b² = c² Using similarity in rt. ∆’s: a² = ce b² = cf a² + b² = ce + cf a² + b² = c(e + f) a² + b² = c(c) = c² Y c a X Z b e c a f d b

Find the length of the leg of the right triangle Find the length of the leg of the right triangle. The hypotenuse is 14 and the other leg is 7. Tell whether the side lengths form a Pythagorean triple.

Special Right Triangles Right triangles whose angle measures are 45 – 45 – 90 or 30 – 60 – 90 are called special right triangles. Angles: 30 - 60 - 90 Sides: x - x√3 - 2x Angles: 45 - 45 - 90 Sides: x - x - x√2 H = 2(shorter leg) L leg = √3 ∙ S leg L leg = √2 ∙ S leg

Formulas 45 – 45 – 90 Triangle Hypotenuse = √2 ∙ leg 30 – 60 – 90 Triangle Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

45 – 45 – 90 Triangle A 3 ? 1/2 B 6√2 5√3 C 5√2 8√6 12 9 C A 45° B

30 – 60 – 90 Triangle A 10 ? 6√2 B 5√3 12 15 C 24 7√3 2√2 C A 60° 30°

Examples x 8√3 X 1) 45° 2) 3) X 12√2 X X X 5√2 4) 5) 6) x x 9 X 10 60° 30° x x 45° 45° 60° 9 X

Trigonometric Ratios and Solving Right Triangles A trigonometric ratio is a ratio of the length of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and the means of measurement of triangles The three basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan) Let ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. SOH CAH TOA Sin A = side opposite A = a hypotenuse c Cos A= side adjacent to A = b hypotenuse Tan A= side opposite A side adjacent to A B hypotenuse a side opposite c c a A = b b C A Side adjacent to A

Examples sin A = ____ 1) sin B =____ cos A =____ cos B =____ 5 Compare the sine and cosine ratios for Angle A and B in the triangle below. B sin A = ____ sin B =____ cos A =____ cos B =____ What do you notice about the sin A and cos B?____ 1) 13 5 C 12 A Find the sine and cosine of angle A and B. Sin A =____ Sin B =____ Cos A =____ Cos B =____ B 3) 18√2 18 C A 18

Given that sin 57 ≈ 0. 839, write the cosine of a complementary angle Given that sin 57 ≈ 0.839, write the cosine of a complementary angle. cos ___ ≈ ____ Given that cos 60 = 0.5, write the sine of a complementary angle. sin ___ = ____

1) Find the sine, cosine, and tangent ratios of Angle A. Example 1) Find the sine, cosine, and tangent ratios of Angle A. B 10 5 C 5√ 3 A

Solving Right Triangles To solve a right triangle means to determine the measures of all six parts. You can solve a right triangle if you know either of the following: Two sides lengths OR One side length and one acute angle measure

Solving Right Triangles Round to the decimals to the nearest tenth Solve the right triangle. Round decimals to the nearest tenth B S 10 15 8 r C A 20° b T R s A Solve the right triangle. Round decimals to the nearest tenth. 12 b B C 10

Angles of Elevation and Depression Looking down Angle of depression Angle of elevation Looking up

Examples: 1. 2.

3. 4. x

5. 6. x 63 12 w z 45 30 160

Word Examples You are measuring the height of a building. You stand 100ft. from the base of the building. You measure the angle of the elevation from a point on the ground to the top of the building to be 48°. Estimate the height of the building. A driveway rises 12ft. Over a distance d at an angle of 3.5°. Estimate the length of the driveway.

Lindsey is 9.2 meters up, and the angle of depression from Lindsey to Pete is 79 degrees. Find the distance from Pete to the base of the building to the nearest tenth of a meter.

Examples During the flight, a hot air balloon is observed by two persons standing at points A and B as illustrated in the diagram. The angle of elevation of point A is 28˚. Point A is 1.8 miles from the balloon as measured along the ground. 0.96 miles 43.8˚ Answer these questions: What is the height h of the balloon? Point B is 2.8 miles from point A. Find the angle of elevation of point B. h A B

The angle of elevation of an antenna is 43° as shown in the diagram. Exercise Problem The angle of elevation of an antenna is 43° as shown in the diagram. If the distance along the ground is 36 feet, find the height of the antenna. Support wires are attached to the antenna as shown with an angle of elevation of 29˚. Find the distance d from the bottom of the antenna to the point where the wires are attached. antenna Support wires 43˚ d 33.6 ft About 20 feet.

8.3 Law of Sines ASA (1 triangle), AAS (1 triangle), SSA (0, 1, or 2 triangles) Use when the triangles are acute or obtuse. sin A sin B sin C a b c = =

Solve the triangles (∆ABC) < A = 45˚, < B = 60˚, a = 14 < C = 88˚, b = 7, c = 7 < B = 130˚, b = 15, c = 11 Example on page 363

8.4 Law of Cosines SAS, SSS (1 triangle) Use when the triangles are acute or obtuse. Formulas on page 368. Words for all 3 are below. 2 2 2 Side Adjacent To angle Other side Adjacent To angle One Adjacent side Other Adjacent side Side Opposite angle Cos (angle) - 2 = +

Solve the triangle. a = 8, b = 5, <C = 60˚ t = 16, s = 14. < R = 120˚ x = 9, y = 40, z = 41

8.5 Problem Solving with Trigonometry Use Trigonometry to Find Triangle Area Formulas on page 376 (SAS) K = ½ (one side)(another side)(sine of included angles)

Proof of Area Trig formula: area = ½ bh = ½ b(c sin A) = ½ bc sin A (one of three formulas) In ∆ABD, Sin A = h/c So h = c sin A. B a c h A C b