6. In each of the triangles drawn for Ex. 5 letter the right angle C, and the acute angles A and B. Measure each side as accurately as possible. Then find to two decimal places the following ratios for each of the triangles: AC BC AC BC

AB

AB BC AC

Compare the corresponding ratios

of the two triangles. What did you and the others of the class find? What is your conclusion?

B

175. Ratios of Angles.-Surveyors, engineers, and scientists in general make daily application of the principles we shall study here. The ratio of any pair of sides of any of the right triangles in the figure equals the ratio of the same pair of sides in any of the other triangles. This can easily be verified by measurements, as was done in Ex. 6 above. Six of these ratios have been given definite names.

commonly used are:

side opposite
hypotenuse

side adjacent
hypotenuse

A

=

EF G Cc

The four most

AB'

The abbreviations, sin A, cos A, tan A, and cot A, are always used. As the angle A increases these ratios will change; sine and tangent will increase, while cosine and cotangent will decrease. Show this. Carefully computed values of these ratios will be found on page 231. The use of these ratios is illustrated in the following problem.

In finding the distance across a pond a surveyor first drives stakes at A and C. He then lays off the line, BC, perpendicular to AC.. He next mea

sures off a distance, BC, equal to a given length and measures the angle, CBA. The right angle at C and the angle, CBA, were both measured by a transit, a picture of which is shown at the bottom of the page.

B

Suppose that the surveyor laid off the line, BC, 240 ft., and that he found the angle, CBA, to be 31°. Then, from the above figure,

Supplying the value of BC and of tan 31° (see page 231),

In making constructions calling for an angle of a given size, use the protractor, as was learned last year.

13.

rately.

14.

15.

Draw three or four angles and measure each accu

Draw an angle of 30°; of 43°; of 76°; of 90°.

Draw a right triangle having one angle 60°. What is the other acute angle?

16. Measure very accurately the sides of the triangle constructed for Ex. 15. Compute the sine, cosine, tangent, and cotangent, of 60°. Compare with the values in the tables. Do you see any connection between the ratios for 60° and those for 30°?

17. Construct a right triangle with one acute angle 56°. Compute the ratios asked for in Ex. 16. Compare with the values in the table.

18. Suppose that it is necessary to find the side, AC, in the accompanying figure. What ratio of the angle, AC CBA, is BC

? Since BC is 8 ft., then

32°

B

Complete the equation and solve for AC. How can AB be found? Find AB.

19. A railroad track rises 1 ft. for every 10 ft. along the track horizontally. The per cent of rise is called the gradient of the track. What is the gradient of this track? The angle the track makes with the horizontal is called the gradient angle. Make a drawing to show the gradient angle. What is the tangent of the gradient angle? Use this to find the gradient angle from the tables. Measure the angle from your drawing to check.

20. A surveyor needing to find the distance across a river sets his transit at N. He next selects some object across the river, as at M, and thinks of an imaginary line, MN, connecting M to N. With his transit he finds the direction of the line, NK, making a right angle with the imaginary line, MN. A definite distance is now measured off on this line, as NK. The transit is now moved to K and the angle, NKM, measured. What ratio of the angle, MN

NKM, is ? If we know the length of NK, how can NK

we find the length of MN?

21. Suppose that angle, MKN, is 26° and the length of NK is 250 ft. What is the width of the river?

22. If angle MKN is 37° and the line, NK, 340 ft., what is the width of the river?

23. One of the perpendicular sides of a right triangle is

35 ft. and the angle opposite is 42°. Find the lengths of the remaining two sides. First make a drawing.

24. The boys in manual training in a junior high school made a large protractor and placed it upon a standard 51 ft. high. A group of the pupils of the school are shown in the picture, using the protractor to measure the height of a building. They placed the standard 27 ft. from the building. They found that they could see the top of the building along a line on the protractor making an angle of 69° with the

of the angle, MHN? Hence, find the value of MN. Also

find the height of the build

ing.

25. Four groups of the same class measured a flagpole to be 58 ft., 57 ft., 55 ft., and 59 ft. They used the average as the real height. What is this?

26. If possible, make a large protractor and place it upon a standard. Use this to measure some high objects as these children did. Take the mean of several results for the final value of the height.

Hold a number contest on finding the ratios of angles, and finding angles having given ratics.