Problem 94 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that for sine, cosine, and tangent, the trig function for the sum of two angles is not equal to that trig function of the first angle plus that trig function of the second angle.

Step-by-Step Solution

Answer

The statement does make sense. The trigonometric function of the sum of two angles is not equal to the sum of the trigonometric function of each angle separately. This is consistent with the trigonometric identities.

Step 1: Understanding the statement

This is a basic observation that is being made about trigonometric functions, specifically sine, cosine, and tangent. The statement is essentially saying that for these functions, the trigonometric function of the sum of two angles (A + B) is not equal to the sum of the trigonometric function of each angle separately (f(A) + f(B)).

Step 2: Analyzing the function properties

There are known identities in trigonometry for the sine, cosine, and tangent of the sum of two angles A and B. They are expressed as follows: \(\sin(A+B) = \sin A \cos B + \cos A \sin B\), \(\cos(A+B) = \cos A \cos B - \sin A \sin B\) and \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). None of these are equal to \(\sin A + \sin B\), \(\cos A + \cos B\) or \(\tan A + \tan B\) respectively. These identities confirm the observation made in the statement. It is interesting to note that this is a direct consequence of the periodic nature of these functions and their behavior. The sum of functions does not retain the cyclic properties of individual trigonometric functions, and it is seen that these properties are instead expressed in combinations of other function products.

Step 3: Clarifying the conclusion

Using the identities discussed, it is clear that the statement makes sense. The sum of the sine, cosine, or tangent of two angles A and B is not equal to the sine, cosine, or tangent of the sum of those two angles (A + B). This is a fundamental property of these functions in trigonometry.