Find the Value

**magentarita** · August 21st 2008, 02:58 AM

Let t = theta for short

(1) If cos(t) = 0.2, find the value of:
cos(t) + cos(t + 2pi) + cos(t + 4pi)

(2) If cot(t) = -2, find the value of:
cot(t) + cot(t - pi) + cot(t - 2pi)

**Sean12345** · August 21st 2008, 03:17 AM

Hi magentarita,

Note that $\cos (x)~\equiv~\cos (x+2k\pi)~\forall k \in \mathbb{Z}$ and that $\cos (x-\pi)~\equiv~-\cos (x)$

**Soroban** · August 21st 2008, 06:25 AM

Hello, magentarita!

Another approach . . .

(1) If $\cos\theta = 0.2$ , find the value of: . $\cos\theta + \cos(\theta + 2\pi) + \cos(\theta + 4\pi)$

Identity: . $\cos(A + B) \;=\;\cos(A)\cos(B) - \sin(A)\sin(B)$

We have: . $\cos\theta + \cos(\theta + 2\pi) + \cos(\theta+4\pi)$

. . . . . . $= \;\cos\theta + \bigg[\cos\theta\cos2\pi-\sin\theta\sin2\pi\bigg] + \bigg[\cos\theta\cos4\pi - \sin\theta\sin2\pi\bigg]$

. . . . . . $= \;\cos\theta + \bigg[\cos\theta\cdot1 - \sin\theta\cdot0\bigg] + \bigg[\cos\theta\cdot1 - \sin\theta\cdot0\bigg]$

. . . . . . $= \;3\cos\theta \;=\;3(0.2) \;=\;\boxed{0.6}$

(2) If $\cot\theta = -2$ , find the value of: . $\cot\theta + \cot(\theta - \pi) + \cot(\theta - 2\pi)$

Identity: . $\cot(A - B) \;=\;\frac{1}{\tan(A - B)} \;=\;\frac{1 +\tan(A)\tan(B)}{\tan(A) - \tan(B)}$

We have: . $\cot\theta + \cot(\theta-\pi) + \cot(\theta-2\pi)$

. . . . . . $=\;\cot\theta + \bigg[\frac{1 + \tan\theta\tan\pi}{\tan\theta - \tan\pi}\bigg] + \bigg[\frac{1 + \tan\theta\tan2\pi}{\tan\theta - \tan2\pi}\bigg]$

. . . . . . $= \;\cot\theta + \bigg[\frac{1 + \tan\theta\cdot0}{\tan\theta - 0}\bigg] + \bigg[\frac{1 + \tan\theta\cdot0}{\tan\theta - 0}\bigg]$

. . . . . . $= \;\cot\theta + \frac{1}{\tan\theta} + \frac{1}{\tan\theta} \;=\;\cot\theta + \cot\theta + \cot\theta$

. . . . . . $= \;3\cot\theta \;=\;3(-2) \;=\;\boxed{-6}$

**Sean12345** · August 21st 2008, 08:50 AM

Ooops.... I misread cot for cos on question 2, thank you Soroban.

**magentarita** · August 21st 2008, 02:43 PM

I thank both of you, especially Soroban.

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