Example 1 Find the value of $$ an 57^{\circ} $$ by drawing and measurement.

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Example 1 Find the value of $$ 	an 57^{\circ} $$ by drawing and measurement.

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Below is one way to approximate tan 57° using a drawing and some simple measuring tools (a protractor and a ruler):

1. Draw a horizontal line and choose a point A on it. This will be the vertex of your angle.

2. From A, use the protractor to draw a ray that makes a 57° angle with the horizontal line. Label any point on the horizontal (say, B) so that AB represents the “adjacent” side.

3. To simplify the measurement, you can set AB = 1 unit (or any convenient length). Along the ray making 57° from A, mark a point C such that you will obtain a right triangle, with the right angle at point D, as follows:
  a. From point B, drop a perpendicular (using a right-angle tool) up to meet the ray at point C. Let the foot of the perpendicular on AB be D.
  b. The triangle ACD is then a right triangle with the angle at A equal to 57°.

4. In this triangle, side AD is along AB (the adjacent side) and side DC (the opposite side) is drawn perpendicular from B (or D) to the ray AC. Measure the lengths of AD and DC using your ruler. (If AB was chosen to be exactly 1 unit, then AD ≈ 1.)

5. Calculate tan 57° as the ratio:
  tan 57° = (length of side opposite the angle) / (length of side adjacent to the angle) = (length of DC) / (length of AD).

6. For example, if you set AD = 1 unit and on measuring you find DC ≈ 1.54 units, then:
  tan 57° ≈ 1.54.

(Using a calculator or trigonometric tables, one also finds that tan 57° is approximately 1.54.)

This construction and measurement illustrate how one can use geometric drawing methods to approximate the tangent of a given angle.
To approximate tan 57°: 1. Draw a horizontal line and mark point A. 2. Draw a 57° angle from A, marking point B on the horizontal. 3. Set AB to 1 unit. 4. Drop a perpendicular from B to the 57° ray, meeting at point C. 5. Measure the lengths of AD (adjacent) and DC (opposite). 6. Calculate tan 57° = DC / AD. For example, if DC ≈ 1.54 units, then tan 57° ≈ 1.54.

Example 1 Find the value of \( \tan 57^{\circ} \) by drawing and measurement.

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