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Sine and Cosine Ratios

16 34. 30 34. 15 17. sin A = or , sin B = or. 30 34. 15 17. 16 34. cos A = or , cos B = or. 8 17. 8 17. Sine and Cosine Ratios. Lesson 8-4. Lesson Quiz. Use this figure for Exercises 1 and 2. 1. Write the ratios for sin A and sin B .

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Sine and Cosine Ratios

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  1. 16 34 30 34 15 17 sin A = or , sin B = or 30 34 15 17 16 34 cos A = or , cos B = or 8 17 8 17 Sine and Cosine Ratios Lesson 8-4 Lesson Quiz Use this figure for Exercises 1 and 2. 1. Write the ratios for sin A and sin B. 2. Write the ratios for cos A and cos B. Use this figure for Exercises 3 and 4. 21.0 3. Find x to the nearest tenth. 4. Find y to the nearest tenth. 13.6 Use this figure for Exercises 5 and 6. 5. Find x to the nearest degree. 6. Find y to the nearest degree. 44 46 8-5

  2. 1.1 ? 2.5 ? 3.3 ? Angles of Elevation and Depression Lesson 8-5 Check Skills You’ll Need (For help, go to Lesson 6-1.) Refer to rectangle ABCD to complete the statements. 4. m1 + m5 ? 5.m10 + m3 ?6. 10 ? Check Skills You’ll Need 8-5

  3. Angles of Elevation and Depression Lesson 8-5 Check Skills You’ll Need Solutions 1. Since all rectangles are parallelograms, AB || DC. By the Alternate Interior Angles Theorem, 1 7. 2. Since all rectangles are parallelograms, AD || BC. By the Alternate Interior Angles Theorem, 5 11. 3. Since all rectangles are parallelograms, AB || DC. By the Alternate Interior Angles Theorem, 3 6. 4. Since ABCD is a rectangle, each of its angles measures 90. So, BDC is a right triangle with mC = 90. The sum of the angles of a triangle is 180, so m 1 + m 5 + m C = 180; substitute for mC:m 1 + m 5 + 90 = 180; subtract 90 from each side: m 1 + m 5 = 90. 5. 10 and 3 form a straight angle so their sum is 180. 6. Since all rectangles are parallelograms, AB || DC. By the Alternate Interior Angles Theorem, 10 8. 8-5

  4. Angles of Elevation and Depression Lesson 8-5 Notes The degree above a horizontal line is the angle of elevation. The degree below a horizontal line is the angle of depression. 8-5

  5. Angles of Elevation and Depression Lesson 8-5 Notes Surveyors use two instruments, the transit and the theodolite, to measure angles of elevation and depression. On both instruments, the surveyor sets the horizon line perpendicular to the direction of gravity. Using gravity to find the horizon line ensures accurate measures even on sloping surfaces. 8-5

  6. One side of the angle of depression is a horizontal line. 1 is the angle of depression from the airplane to the building. One side of the angle of elevation is a horizontal line. 2 is the angle of elevation from the building to the airplane. Angles of Elevation and Depression Lesson 8-5 Additional Examples Identifying Angles of Elevation and Depression Describe 1 and 2 as they relate to the situation shown. Quick Check 8-5

  7. Draw a diagram to represent the situation. x 200 tan 35° = Use the tangent ratio. Solve for x. x = 200 • tan 35° Use a calculator. 200 35 140.041508 So x 140. Angles of Elevation and Depression Lesson 8-5 Additional Examples Real-World Connection A surveyor stands 200 ft from a building to measure its height with a 5-ft tall theodolite. The angle of elevation to the top of the building is 35°. How tall is the building? To find the height of the building, add the height of the Theodolite, which is 5 ft tall. Quick Check The building is about 140 ft + 5 ft, or 145 ft tall. 8-5

  8. An airplane flying 3500 ft above ground begins a 2° descent to land at an airport. How many miles from the airport is the airplane when it starts its descent? Draw a diagram to represent the situation. 3500 x sin 2° = Use the sine ratio. Use a calculator. 3500 sin 2° 3500 2 100287.9792 x = Solve for x. Divide by 5280 to convert feet to miles. 5280 18.993935 Angles of Elevation and Depression Lesson 8-5 Additional Examples Real-World Connection Quick Check The airplane is about 19 mi from the airport when it starts its descent. 8-5

  9. Angles of Elevation and Depression Lesson 8-5 Lesson Quiz Use the diagram for Exercises 1 and 2. 1. Describe how 1 relates to the situation. 2. Describe how 2 relates to the situation. angle of elevation from man’s eyes to treetop angle of depression from treetop to man’s eyes A 6-ft man stands 12 ft from the base of a tree. The angle of elevation from his eyes to the top of the tree is 76°. 3. About how tall is the tree? 4. If the man releases a pigeon that flies directly to the top of the tree, about how far will it fly? 5. What is the angle of depression from the treetop to the man’s eyes? about 54 ft about 50 ft 76° 8-5

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