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Acute Angles and Right Triangles

Learn to solve triangles by finding angles and sides, understand significant digits in calculations, and apply trigonometry to real-world scenarios. Examples provided for practical understanding.

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Acute Angles and Right Triangles

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  1. 2 Acute Angles and Right Triangles

  2. 2 Acute Angles and Right Triangles • 2.1 Trigonometric Functions of Acute Angles • 2.2 Trigonometric Functions of Non-Acute Angles • 2.3 Finding Trigonometric Function Values Using a Calculator • 2.4 Solving Right Triangles • 2.5 Further Applications of Right Triangles

  3. Solving Right Triangles 2.4 Significant Digits ▪ Solving Triangles ▪ Angles of Elevation or Depression

  4. Significant Digits • A significant digit is a digit obtained by actual measurement. • The significant digits in the following numbers are identified in color. • 408 21.5 18.00 6.700 0.0025 0.09810 7300 • Your answer is no more accurate than the least accurate number in your calculation.

  5. To determine the number of significant digits for answers in applications of angle measure, use the following table.

  6. Solving Triangles • To solve a triangle means to find the measures of all the angles and sides of the triangle. • When solving triangles, a labeled sketch is an important aid. • Use a to represent the length of the side opposite angle A, b for the length of the side opposite angle B, and so on. • In a right triangle, the letter c is reserved for the hypotenuse.

  7. Example 1 SOLVING A RIGHT TRIANGLE GIVEN AN ANGLE AND A SIDE • Solve right triangle ABC, if A = 34°30′ and c = 12.7 in.

  8. Example 1 SOLVING A RIGHT TRIANGLE GIVEN AN ANGLE AND A SIDE (continued) • We could have found the measure of angle B first and then used the trigonometric function values of B to find the unknown sides. A right triangle can usually be solved in several ways, each producing the correct answer. To maintain accuracy, always use given information as much as possible, and avoid rounding off in intermediate steps.

  9. Example 2 SOLVING A RIGHT TRIANGLE GIVEN TWO SIDES • Solve right triangle ABC, if a = 29.43 cm and c = 53.58 cm. or 33º 19΄

  10. Angles of Elevation or Depression • The angle of depression from point X to point Y (below X) is the acute angle formed by ray XY and a horizontal ray with endpoint X. • The angle of elevation from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with endpoint at X.

  11. Caution Be careful when interpreting the angle of depression. Both the angle of elevation and the angle of depression are measured between the line of sight and a horizontal line.

  12. Solving an Applied Trigonometry Problem Step 1 Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. Step 2 Use the sketch to write an equation relating the given quantities to the variable. Step 3 Solve the equation, and check that your answer makes sense.

  13. Example 3 FINDING A LENGTH GIVEN THE ANGLE OF ELEVATION • Pat Porterfield knows that when she stands 123 ft from the base of a flagpole, the angle of elevation to the top of the flagpole is 26°40′. If her eyes are 5.30 ft above the ground, find the height of the flagpole. Since Pat’s eyes are 5.30 ft above the ground, the height of the flagpole is 61.8 + 5.30 = 67.1 ft.

  14. Example 4 FINDING AN ANGLE OF DEPRESSION • From the top of a 210-ft cliff, David observes a lighthouse that is 430 ft offshore. Find the angle of depression from the top of the cliff to the base of the lighthouse. The angle of depression is measured from a horizontal line down to the base of the lighthouse. The angle of depression and angle B, in the right triangle shown, are alternate interior angles whose measures are equal. We use the tangent ratio to solve for angle B.

  15. Example 4 FINDING AN ANGLE OF DEPRESSION (continued)

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