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Chapter 4: Problem 81

Determine whether the statement is true or false. Justify your answer. $$\sin 45^{\circ}+\cos 45^{\circ}=1$$

Short Answer

Expert verified
The statement \(\sin 45^{\circ}+\cos 45^{\circ}=1\) is false.

Step by step solution

01

Recall the values of common angles in trigonometry

In trigonometry, some angles have easily remembered values. 45 degrees is one such case both for sine and cosine. We know that \(\sin 45^{\circ}=\frac{1}{\sqrt{2}}\) and \(\cos 45^{\circ}=\frac{1}{\sqrt{2}}\).
02

Substitute values into the statement

Now we can substitute these values into our statement. Thus, \(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}}\). We can rationalize the denominator by multiplying by \(\frac{\sqrt{2}}{\sqrt{2}}\) to get \(\sqrt{2}\). This shows that our initial statement equals to \(\sqrt{2}\), not 1.
03

Conclusion

Since we have discovered that the left-hand side of the expression equates to \(\sqrt{2}\) and not 1, we can conclude our expression is false.

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