Making the most out of our graphical telemetry computers.

Like with many complex instruments, there are multiple ways to solve certain problems and to measure the required inputs. Combining the various functions leads to a multitude of advanced uses.

## Finding Tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a tangent to the circle (the line that touches the circle), from Latin linea tangens or touching line (tangere, to touch).

With the Telemeter Quadrant, there are two ways of determining the tangent of an angle. Finding tangent with the shadow square. Note that tan(35°) approx. 7/10 = 0.7 (umbra versa). Find the mark where the string crosses the shadow square and determine whether the result lies closer to the 10 or 12-divisions. Divide accordingly.

Now since tangent equals opposite over adjacent, we can use the shadow square by dividing the cosine into tenths or twelfths, in order to easily calculate the tangent as

``````tan(θ) = sin(θ) / cos(θ)
``````

#### Degree to tangent

1. Set string to degrees; 11°
2. Read mark where the string is crossing the shadow square; 2 on the inner scale with 10 divisions
3. Thus tan(11°) = 2/10 = 0.2
4. (Note that for distance calculations, 1/tan(11°) would be calculated as 10/2 = 5)

#### (Arc)tangent to degree

1. Set string to tangent on the shadow square; 3/12
2. Read mark where the string is crossing the degree scale; 14°
3. Thus arctan(3/12) = 14°

Note that cot(x) = 1/tan(x), so cotangent is the reciprocal of a tangent while arctan(x) or tan-1(x) is the inverse tangent which returns the angle whose tangent is x.

### 2. Slope scale

The second method to find the tangent of an angle is to use the slope scale, which is labeled with percent values, where the respective decimal value is the tangent of the angle on the adjacent degree scale. Finding tangent with the slope scale. Find the mark where the string crosses the degree scale and read the tangent on the slope scale.

In our example, we pull the string across the 35° and are able to directly read the tangent of 35° as 70% = 0.7 on the slope scale.

To compute inverse tangent (arctangent), we can use the slope scale to set the value, e.g. 75% (= 0.75) and read the arctan on the degree scale (about 36.5°).