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(c) tan 15 or cot 15 (d) sec 55 or csc 55 (c) cot 15 , (d) sec 55
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Find the exact value of each of the following.
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(a) sin 30 tan 45 (b) cot 45 cos 60 (c) sin 30 cos 60 cos 30 sin 60
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CHAPTER 3 Trigonometric Functions of an Acute Angle
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(d) cos 30 cos 60 sin 30 sin 60 tan 60 tan 30 (e) 1 tan 60 tan 30 csc 30 csc 60 csc 90 (f) sec 0 sec 30 sec 60
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Ans. 3.21
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(a) 3/2,
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(c) 1,
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(d) 0,
(e) 1> 23
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(f) 1
A man drives 500 m along a road which is inclined 20 to the horizontal. How high above his starting point is he Ans. Manual: 170 m; calculator: 171 m (manual answer differs because of rounding of table values).
A tree broken over by the wind forms a right triangle with the ground. If the broken part makes an angle of 50 with the ground and the top of the tree is now 20 ft from its base, how tall was the tree Ans. 55 ft
Two straight roads intersect to form an angle of 75 . Find the shortest distance from one road to a gas station on the other road that is 1000 m from the intersection. Ans. Manual: 3730 m; calculator: 3732 (manual answer differs because of rounding of table values).
Two buildings with flat roofs are 60 m apart. From the roof of the shorter building, 40 m in height, the angle of elevation to the edge of the roof of the taller building is 40 . How high is the taller building Ans. 90 m
A ladder with its foot in the street makes an angle of 30 with the street when its top rests on a building on one side of the street and makes an angle of 40 with the street when its top rests on a building on the other side of the street. If the ladder is 50 ft long, how wide is the street Ans. 82 ft
Find the perimeter of an isosceles triangle whose base is 40 cm and whose base angle is 70 . Ans. 157 cm
Solution of Right Triangles
4.1 Introduction
The solution of right triangles depends on using approximate values for trigonometric functions of acute angles. An important part of the solution is determining the appropriate value to use for a trigonometric function. This part of the solution is different when you are using tables (as in Secs. 4.2 to 4.4) from when you are using a scientific calculator (as in Secs. 4.5 and 4.6.) In general, the procedure will be to use the given data to write an equation using a trigonometric function and then to solve for the unknown value in the equation. The given data will consist either of two sides of a right triangle or of one side and an acute angle. Once one value has been found, a second acute angle and the remaining side can be found. The second acute angle is found using the fact that the acute angles of a right triangle are complementary (add up to 90 ). The third side is found by using a definition of a second trigonometric function or by using the Pythagorean theorem (see App. 1, Geometry).
4.2 Four-Place Tables of Trigonometric Functions
App. 2, Tables, has three different four-decimal-place tables of values for trigonometric functions, with Table 1 giving angles in 10 intervals, Table 2 giving angles in 0.1 intervals, and Table 3 giving angles in 0.01-rad intervals. Tables published in texts differ in several ways, such as in the number of digits listed, the number of decimal places in each value, whether or not secant and cosecant values are listed, and the measurement unit of the angles. The angles in Tables 1 and 2 are listed in the left- and right-hand columns. Angles less than 45 are located in the left-hand column, and the function is read from the top of the page. Angles greater than 45 are located in the right-hand column, and the function is read from the bottom of the page. In each row, the sum of the angles in the left- and right-hand columns is 90 , and the tables are based on the fact that cofunctions of complementary angles are equal. In Table 3, the angles in radians are listed in the left-hand column only, and the function is read from the top of the page.
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