Lesson 4

You are now ready to apply your understanding of how trigonometric identities can be used to solve equations. Go to Module 5 Project: Pushing the Limits of Vehicle Performance. Complete Activity 1: Part 2.

 

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Going Beyond

 

Prosthaphaeresis

 

Before logarithms were used, and much before pocket calculators were invented, product identities were used to estimate the value of products. An example of a product identity is

 

The process is called prosthaphaeresis and can be thought of in four steps:

• Scale down by moving the decimal place of each value so it is between 0 and 1.
• Use cos−1 on each value to determine an A and B value. (Historically this would have been done with a table of cosine values.)
• Use the identity to determine the product.
• Scale up by moving the decimal place to the right (or left) as many places as you moved it to the left (or right) for each input.

Consider 127 × 588.

 

Step 1: 0.127 × 0.588

 

Step 2: = cos (1.4434…) cos (0.9422…)

 

Step 3:

 

Step 4: 74 606

 

So, 127 × 588 = 74 606.