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Chapter 5: Problem 94

Determine whether the statement is true or false. Justify your answer. $$\sin \left(x-\frac{\pi}{2}\right)=-\cos x$$

Short Answer

Expert verified
The statement is true. We proved that \(\sin \left(x - \frac{\pi}{2}\right)\) is indeed equal to \(-\cos(x)\) by using the properties of sine and cosine functions, as well as by substitution of values.

Step by step solution

01

Understand the properties of sin and cos

The key property to remember here is the phase shift of sin(x) into cos(x). In trigonometry, the sin function is identical to the cos function, but shifted to the right by \(\frac{\pi}{2}\) or to the left by \(\frac{-\pi}{2}\). This means that \(\sin \left(x - \frac{\pi}{2}\right)\) should be equivalent to \(\cos(x)\). However, in our expression it says that it should be equivalent to \(-\cos(x)\), not \(\cos(x)\). Let's proceed to substitute x with some values to verify this.
02

Substitution

Let's start by taking some simple known values for x, such as \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \(2\pi\). Then we substitute these values into both the expressions and see what values we obtain.
03

Comparing the results

With the values of x from step 2, substitute those into the expressions. We will find that the results from \(\sin \left(x - \frac{\pi}{2}\right)\) and \(-\cos(x)\) are identical, verifying the given equation.

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Most popular questions from this chapter

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