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# Geometry for Modeling and Design

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## 4.8 Laying Out an Angle

Many angles can be laid out directly with the triangle or protractor. For more accuracy, use one of the methods shown in Figure 4.31.

Tangent Method The tangent of angle θ is y/x, and y = x tan θ. Use a convenient value for x, preferably 10 units (Figure 4.31a). (The larger the unit, the more accurate will be the construction.) Look up the tangent of angle θ and multiply by 10, and measure y = 10 tan θ.

• EXAMPLE To set off 31-1/2°, find the natural tangent of 31-1/2°, which is 0.6128. Then, y = 10 units × 0.6128 = 6.128 units.

Sine Method Draw line x to any convenient length, preferably 10 units (Figure 4.31b). Find the sine of angle θ, multiply by 10, and draw arc with radius R = 10 sin θ. Draw the other side of the angle tangent to the arc, as shown.

• EXAMPLE To set off 25-1/2°, find the natural sine of 25-1/2°, which is 0.4305. Then R = 10 units × 0.4305 = 4.305 units.

Chord Method Draw line x of any convenient length, and draw an arc with any convenient radius R—say 10 units (Figure 4.31c). Find the chordal length C using the formula C = 2 sin θ/2. Machinists’ handbooks have chord tables. These tables are made using a radius of 1 unit, so it is easy to scale by multiplying the table values by the actual radius used.

• EXAMPLE Half of 43°20′ = 21°40′. The sine of 21°40′ = 0.3692. C = 2 × 0.3692 = 0.7384 for a 1 unit radius. For a 10 unit radius, C = 7.384 units.

• EXAMPLE To set off 43°20′, the chordal length C for 1 unit radius, as given in a table of chords, equals 0.7384. If R = 10 units, then C = 7.384 units.