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Chapter 4: Problem 84

Determine whether the statement is true or false. Justify your answer. $$\tan \left[\left(5^{\circ}\right)^{2}\right]=\tan ^{2} 5^{\circ}$$

Short Answer

Expert verified
The statement \(\tan \left[(5^{\circ})^{2}\right]=\tan^{2} 5^{\circ}\) is False.

Step by step solution

01

Simplify the expression on the left side of the equation

Recalling the order of operations (brackets, powers, division and multiplication, addition and subtraction, BDMPAS), the left side \( \tan \left[(5^{\circ})^{2}\right] \) simplifies first by squaring 5 degrees and then taking the tangent of that result. So, \[ \tan \left[(5^{\circ})^{2}\right] = \tan \left[ (25)^{\circ}\right] \]. Using a calculator to find \(\tan 25^{\circ}\), we get approximately 0.4663.
02

Simplify the expression on the right side of the equation

On the other hand, the right side simplifies differently. Here, \(\tan^{2} 5^{\circ}\) means the tan of 5 degrees is being squared. That is, \(\tan^{2} 5^{\circ} = \left[\tan \left(5^{\circ}\right) \right]^{2}\). Using a calculator to find \(\tan 5^{\circ}\), we get approximately 0.0875. Squaring this gives approximately 0.0076.
03

Comparing the simplified expressions

Comparing the results of the left side (approximately 0.4663) and right side (approximately 0.0076), it's observable that they aren't equal.

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