MHB How can the given equation be verified without using a calculator?

[mathdad]

The instructions are as follows:

Use your calculator to verify the given equation.

cos (1 + pi/2) = - sin 1

I was easily able to do this with my calculator. My question, however, is: how can I verify the equation without my calculator?

 

[Dethrone]

It comes right from the identity $\cos\left({\theta+\pi/2}\right)=-\sin\left({\theta}\right)$, or just think about it geometrically. Draw a cosine shifted to the left by $\pi/2$ and compare it with a regular sine wave. Or, apply the cosine addition formulas if you're familiar with them.

 

[Greg]

Another approach:

Use the identity

$$\cos(\theta\pm\varphi)=\cos(\theta)\cos(\varphi)\mp\sin(\theta)\sin(\varphi)$$

and the facts that

$$\cos\left(\frac{\pi}{2}\right)=0,\quad\sin\left(\frac{\pi}{2}\right)=1$$

 

[mathdad]

Another approach:

Use the identity

$$\cos(\theta\pm\varphi)=\cos(\theta)\cos(\varphi)\mp\sin(\theta)\sin(\varphi)$$

and the facts that

$$\cos\left(\frac{\pi}{2}\right)=0,\quad\sin\left(\frac{\pi}{2}\right)=1$$

I am familiar with this identity even though it is several chapters away in my textbook. I took a class at NYC Technical College in the late 1980s called Algebra 2 and Trigonometry. We used this formula quite a bit in that class aka MA185.

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It comes right from the identity $\cos\left({\theta+\pi/2}\right)=-\sin\left({\theta}\right)$, or just think about it geometrically. Draw a cosine shifted to the left by $\pi/2$ and compare it with a regular sine wave. Or, apply the cosine addition formulas if you're familiar with them.

Thank you but I think using the trig identity is a lot easier than graphing cosine or any of the other trig functions. I do not recall the last time I had to graph a trig function by hand. For all graphs, I just use mathway.com or wolfram.