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Ratios in Right Triangles

Ratios in Right Triangles. Expectations: G1.3.1: Define and use sine, cosine and tangent ratios to solve problems using trigonometric ratios in right triangles. Determine the exact values of sine, cosine and tangent for various angle measures. Daily Quiz 5/9/2011.

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Ratios in Right Triangles

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  1. Ratios in Right Triangles Expectations: G1.3.1: Define and use sine, cosine and tangent ratios to solve problems using trigonometric ratios in right triangles. Determine the exact values of sine, cosine and tangent for various angle measures. 8-3: Ratios in Right Triangles

  2. Daily Quiz 5/9/2011 • An isosceles right triangle has a leg of length 7 inches. Determine the perimeter of the right triangle?

  3. BC is the leg opposite A Opposite Legs • From an acute angle in a right triangle, the leg opposite is the leg that lies in the interior of the angle (except the endpoints of the side). B C A 8-3: Ratios in Right Triangles

  4. AC is the leg opposite ∠B Opposite Legs • From an acute angle in a right triangle, the leg opposite is the leg that lies in the interior of the angle (except the endpoints of the side). B C A 8-3: Ratios in Right Triangles

  5. AC is the leg adjacent ∠A Adjacent Legs • The leg adjacent to an acute angle of a right triangle is the leg that forms a side of the acute angle. B C A 8-3: Ratios in Right Triangles

  6. BC is the leg adjacent ∠B Adjacent Legs • The leg adjacent to an acute angle of a right triangle is the leg that forms a side of the acute angle. B C A 8-3: Ratios in Right Triangles

  7. Sine Ratio • The sine ratio of an acute angle of a right triangle compares the length of the leg opposite the angle to the length of the hypotenuse. • Sine is abbreviated sin, but it is still read as “sine”. 8-3: Ratios in Right Triangles

  8. Sine Ratio B C A leg opposite sin θ = hypotenuse 8-3: Ratios in Right Triangles

  9. sin A = sin B = Sine Ratio B C A AC BC AB AB 8-3: Ratios in Right Triangles

  10. Cosine Ratio • The cosine ratio of an acute angle in a right triangle compares the length of the leg adjacent the acute angle to the length of the hypotenuse. • Cosine is abbreviated “cos” but is still read as “cosine.” 8-3: Ratios in Right Triangles

  11. Cosine Ratio B C A leg adjacent cos θ = hypotenuse 8-3: Ratios in Right Triangles

  12. cos A = cos B = Cosine Ratio B C A BC AC AB AB 8-3: Ratios in Right Triangles

  13. B 10 6 C A 8 Give the sin and cos ratios for ∠A and ∠B. 8-3: Ratios in Right Triangles

  14. B b a c A C For the right triangle shown below, what is the sin C? • a/b • a/c • b/a • c/b • c/a

  15. Daily Quiz May 10, 2011 • An isosceles right triangle and a 30-60-90 right triangle both have shortest side of length 4. Draw the 2 triangles and determine the exact lengths of all of the sides.

  16. Solve for x in the triangle below. 24 35° x 8-3: Ratios in Right Triangles

  17. Solve for x in the triangle below. x 75° 18 8-3: Ratios in Right Triangles

  18. Daily Quiz 5/11/2011If AC = 10 in the figure below, determine BD. D B 30° 45° A C Trig Basics

  19. Tangent Ratio • The tangent ratio of an acute angle of a right triangle compares the length of the leg opposite the acute angle to the length of the leg adjacent the acute angle. • Tangent is abbreviated “tan” but is still read as “tangent.” 8-3: Ratios in Right Triangles

  20. Tangent Ratio B C A leg opposite tan θ = leg adjacent 8-3: Ratios in Right Triangles

  21. tan A = tan B = Tangent Ratio B C A AC BC BC AC 8-3: Ratios in Right Triangles

  22. Tangent Ratio Solve for x in the triangle below. 15 65° x 8-3: Ratios in Right Triangles

  23. Daily Quiz 5/13/11 • Draw a right triangle with an acute angle of 40 degrees. If the leg opposite the 40 degree angle has length of 7 inches, determine the length of the hypotenuse.

  24. A kite is flying at the end of a 240-foot string which makes a angle with the horizon. If the hand of the person flying the kit is 3 feet above the ground, how far above the ground is the kite? Trig Basics

  25. Arc functions • If you know the value of a trig function, you can work backwards to determine the measure of the angle. • For example, say we know the cos A = .5, then we can use the cos-1 (arc cosine or inverse of cosine) function to determine that m∠A = 60°. 8-3: Ratios in Right Triangles

  26. To calculate angles from cos: • Use the 2nd (shift or inverse) key before the cos key. • Ex: cos A = .8894 • Type .8894 2nd cos . • This returns 27.20, so m∠ A = 27.2° • You may need to type 2nd cos .8894 = 8-3: Ratios in Right Triangles

  27. To calculate angles from sin: • Use the 2nd (shift or inverse) key before the sin key. • Ex: sin A = .6 • Type .6 2nd sin . • This returns 36.87, so m∠A = 36.87° • You may need to type 2nd sin .6 = 8-3: Ratios in Right Triangles

  28. To calculate angles from tan: • Use the 2nd (shift or inverse) key before the tan key. • Ex: tan A = .2341 • Type .2341 2nd tan . • This returns 13.17, so m∠ A = 13.17° • You may need to type 2nd tan .2341 = 8-3: Ratios in Right Triangles

  29. A right triangle has sides of length 5, 12 and 13 cm. Determine the measure of both of the acute angles in the triangle.

  30. A patient is being treated with radiotherapy for a tumor that is behind a vital organ. In order to prevent damage to the organ, the doctor must angle the rays to the tumor. If the tumor is 6.3 cm below the skin and the rays enter the body 9.8 cm to the right of the tumor, find the angle at which the rays should enter the body to hit the tumor. 8-3: Ratios in Right Triangles

  31. Daily Quiz 5/13 • To guard against a fall, a ladder should form no more than a 75° angle with the ground. What is the maximum height that a 10 foot ladder can safely reach? 8-3: Ratios in Right Triangles

  32. The hypotenuse of the right triangle shown below is 22 feet long. The cosine of angle L is ¾. How many feet long is the segment LM? • 18.4 • 16.5 • 11.0 • 6.7 • 4.7 L 22 M N 8-3: Ratios in Right Triangles

  33. Solve for x and y below. y 12 22° x 8-3: Ratios in Right Triangles

  34. The side lengths of a right triangle are 3, 4 and 5 inches. What are the measures of the acute angles?

  35. Assignment • Pages 416 – 419, • # 17-49 (odds), 50, 51, 53 – 65 (odds) 8-3: Ratios in Right Triangles

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