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table on page 141 gives the tangent ratios* for acute angles from 0° to 89°. Thus for an angle of 30° we find the tangent ratio to be .577. Briefly, we say the tangent of 30° is .58, which may be written

tan 30° = .58

EXERCISES

1. From the table of tangents find the tangent ratios of the following angles: 10°; 22°; 45°; 67°; 82°.

In each case state your result in the form of an equation, as tan 10°.176.

2. Find the angles corresponding to the following tangent ratios: 57 State results in the form of equa4'8

.141; .306; .601; 1.73; 6.31;

tions.

96. How to use the right-triangle method. Exercise

1, below, shows how to find unknown distances by

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at A is 65° approximately. How high is the balloon?

*There are other ratios, e. g., the opposite side to the hypotenuse, and the adjacent side to the hypotenuse. The table of tangents is sufficient for the solution of the problems which follow.

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Multiplying both members of the equation by 181,

h=(181) X (2.145)=388.245.

Since in the product | Computation:

2.145×181,

=

2145 first partial product, second partial product

17160

2145

(181) (2.145) the 5 in the 2.145 and the 1 in 181 are uncertain, the first partial product 2145 is uncertain. Furthermore, in the other partial products, the 8X5 and 1X5 are uncertain. Hence, in the sum of the partial products the last four figures are uncertain.

=third partial product

388.245 product.

=

All the doubtful numbers in the computation above have been marked with dots placed over the number. Since the second 8 in 388 is doubtful, the figures following are meaningless, and should be dropped.

.. The balloon is about 388 ft. high.

2. Make a summary of the steps in the solution of Exercise 1.

3. The rope of a flagpole is stretched so that it touches the ground at a point 18 ft. from the foot of the pole. The rope makes an angle of 70° with the ground. Find the approximate height of the pole, writing your computation as shown in Exercise 1.

4. A pole 22 ft. high casts a shadow 16 ft. long. Find the angle of elevation of the sun.

Solution: Let x be the required angle.

22

tan x=

16

Computation:
1.37

We may assume that in the measurement of the pole and the shadow the last figures in 22 and 16 were approximate. The division shows 16)22 that in the quotient the first figure to the right of the decimal is doubtful and the second meaningless.

Hence tan x=1.4, and from the table

x=54° approximately.

16

6.0

4.8

120

112

5. A vertical pole 9 ft. long casts a shadow, on level ground, 11 ft. long. Find the angle of elevation of the sun.

21⁄2 ft.

25 ft. FIG. 179

6. What is the angle of elevation of a road which rises 2 ft. in a horizontal distance of 25 feet (Fig. 179)?

7. In the triangle (Fig. 180) find the approximate values of angle x for the following values of a and b.

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8. Find the value of a (Fig. 180) corresponding to the value of x and b given in the following table:

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97. What every pupil should know and be able to do. Having studied Chapter V every pupil should be able to do the following:

1. To find unknown distances by the methods taught in the chapter.

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2. To solve for the unknown equations of the type

16

3 5

3. To determine the degree of accuracy that can be obtained in the product of two decimal fractions in which the last figure to the right is doubtful.

4. To use compass, protractor, and squared paper in drawing triangles and angles, and measuring segments. 5. To solve some simple verbal problems by means of proportions.

The following principles should be known:

1. Two triangles are congruent, if two sides and the included angle of one are equal to two sides and the included angle of the other.

2. Two triangles are congruent if two angles and the side included between their vertices in one triangle are equal respectively to the corresponding parts of the other.

3. If the angles of one triangle are equal to the angles of another, the triangles are similar.

4. If the corresponding angles of two triangles are equal, the triangles are similar, and the ratios of the corresponding sides are equal.

5. In a proportion the product of the means is equal to the product of the extremes.

6. If equal numbers are multiplied by the same, or equal, numbers the products are equal (multiplication axiom).

7. The pupil should be familiar with the meaning of the following terms: congruent polygons, scale drawing, angle of elevation, angle of depression, similar polygons, proportion, tangent ratio.

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gruent with the first. Draw a third triangle similar to the first.

4. Find the distance AB (Fig. 183) by means of a scale drawing.

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11. By means of a proportion, find the tax on a property assessed at $18,000, if a tax of $72.40 is paid on property assessed at $3280.

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