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Trig Identifies and Equations: Tangent Sum Formulas and Tire Circumferences

1. Two cars with new tires are driven at an average speed of 60 mph for a test drive of 2000 miles. The diameter of the wheels of one car is 15 inches. The diameter of the wheels of the other car is 16 inches. If the tires are equally durable and differ only by diameter, which car will probably need new tires first? Why?

2. Explain why tan(x + 450 degrees) cannot be simplified using the tangent sum formulas but can be simplified by using the sine and cosine formulas.

3. What is the difference between a trigonometric equation that is an identity and a trigonometric equation that is not an identity? Provide an example to clarify.

Solution Preview

The car whose tires (particularly, any given point on the tires) made contact with the road more than the other will be the car that will probably need new tires first. It is contact with the road that wears down the tires. Therefore, the tires (particularly, at any given point on the tires) with the smallest diameter made contact with the road more. Therefore, the car with the 15-inch diameter tires will need tires first.

Another way to look at it is this. Consider, what is the circumference of each tire?

The 15-inch tire has a circumference of (pi x diameter) = 47 in
The 16-inch ...

Solution Summary

In this solution, I have provided easy to understand and step-by-step explanations for these three trigonometric based questions. In total, this response is just under 450 words. All required calculations are provided.

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