SECTION 8.1 Right Triangle Trigonometry; Applications 549 To solve a right triangle, we need to know one of the acute angles A or B and a side, or else two sides (in which case the Pythagorean Theorem can be used). Also, because the sum of the measures of the angles of a triangle is 180 ,° the sum of the measures of angles A and B in a right triangle must be 90 .° Figure 7 Right triangle c a b B A Figure 8 408 c B 2 a TIP To avoid round-off errors when using a calculator, we will store unrounded values in memory for use in subsequent calculations. j Figure 9 c A B 3 2 THEOREM Properties of a Right Triangle For the right triangle shown in Figure 7 on the previous page and repeated here, we have c a b A B 90 2 2 2 = + + = ° Solving a Right Triangle Use Figure 8. If b 2 = and A 40 , = ° find a, c, and B. Solution EXAMPLE 4 Because A 40 = ° and A B 90 , + = ° it follows that B 50 . = ° To find the sides a and c, use the facts that a c tan40 2 and cos40 2 ° = ° = Now solve for a and c. = ° ≈ = ° ≈ a c 2 tan40 1.68 and 2 cos40 2.61 Now Work PROBLEM 2 9 Solving a Right Triangle Use Figure 9. If a 3 = and b 2, = find c, A, and B. Solution EXAMPLE 5 Because a 3 = and b 2, = then, by the Pythagorean Theorem, c a b c 3 2 9 4 13 13 3.61 2 2 2 2 2 = + = + = + = = ≈ To find angle A, use the fact that A A tan 3 2 so tan 3 2 1 = = − Use a calculator with the mode set to degrees to find that A 56.3 = ° rounded to one decimal place. Since A B 90 , + = ° this means that B 33.7 . = ° 4 Solve Applied Problems * In addition to developing models using right triangles, we can use right triangle trigonometry to measure heights and distances that are either awkward or impossible to measure by ordinary means. When using right triangles to solve these problems, pay attention to the known measures. This will indicate what trigonometric function to use. For example, if you know the measure of an angle and the length of the side adjacent to the angle, and wish to find the length of the opposite side, you would use the tangent function. Do you know why? *In applied problems, it is important that answers be reported with both justifiable accuracy and appropriate significant figures. In this chapter we shall assume that the problem data are accurate to the number of significant digits resulting in sides being rounded to two decimal places and angles being rounded to one decimal place. Now Work PROBLEM 3 9
RkJQdWJsaXNoZXIy NjM5ODQ=