TRIGONOMETRIC FUNCTIONS

Tangents of Angles from 0°34′ to 5°45′
The values of tangents and sines of angels smaller than 5°45′ are so nearly alike that they may be considered identical for slide rule computations. Consequently, for angles from 0°34′ to 5°45′, tangents can be read directly using the S scale and the left half of the A or B scales, as described in the section of sines.
Tangents of Angles greater than 45°
Using the relationship
 tan x = 1 , tan (90 – x)
tangents for angles greater than 45° can be directly computed on the slide rule.
Example: Find tan 69°.
Solution: 90 – 69 = 21. Set 21 on the T scale under the right-hand fixed hairline. Example shown in Figure 25a. The answer, which is the reciprocal of tan 21, is read directly on the D scale under the left index of the C scale as 2.605. Example shown in Figure 25b. Note that for angles between 45° and approximately 84° the value of the tangent ranges from 1 to 10.
When in the above relationship 90 – x is less than 5°45′, the left half of the S scale is used in making the computation. As pointed out above, the sines and tangents of small angles may be considered equal for slide rule purposes.
Example: Find tan 88°.
Set 2 on the S scale (90 – 88) under the right-hand fixed hairline. Example shown in Figure 26a. Turn the slide rule over and read the answer 28.6 on the A scale over the left index of the B scale. Example shown in Figure 26b. Using this method, tangents of angles up to 89°25′ can be read directly.
Sines and Tangents of Very Small Angles
For determining the sine or tangent of a very small angle, two gauge points are provided on the S scale. The one identified by the symbol (′) is called the minutes gauge point and is fount next to the 2° mark on the S scale. The second gauge point marked (″) is found near the 1°10′ mark on the S scale. Example shown in Figure 27. They represent the value of the angle in radians. Their use is based on the fact that for very small angles, sin x = tan x = x (in radians), approximately.
Example: Find sin 3′.
Since sin 3′ = 3′ (in radians), solve 3 × 1′ (in radians). The procedure when using these gauge points is the same as the use of the CI and D scale combination for multiplication. With the slide rule turned so that the trig scales are face up, set the minutes gauge mark under the 3 on the A scale. Over the right index on the S scale, read 0.000873 on the A scale. Example shown in Figure 28.
Example: Find sin 3″.
Set the second gauge mark under 3 on the A scale. Over the left index on the S scale, read 0.0000145 on the A scale. Example shown in Figure 29.
The decimal point in the above examples is located by noting that: