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sergij07 [2.7K]

2 years ago

12

QUICK HELP! How many terms are in the simplified expression? How many of the terms in the simplified expression are negative?

1 answer:

Bess [88]2 years ago

3 0

There are 4 terms.
The second term and the fourth term have negative coefficients.
Terms are only negative depending on the value of the variables.

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If tan^3(theta) -1/tan(theta)-1 - sec^2(theta) + 1 = 0 find cot(theta)

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$\tan^3\theta-\dfrac1{\tan\theta-1}-\sec^2\theta+1=0$

Recall that $\tan^2\theta+1=\sec^2\theta$ , so we can write everything in terms of $\tan\theta$ :

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Let $x=\tan\theta$ , so that

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With some rewriting we get

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The numerator determines when the expression on the left reduces to 0:

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$x^2(x-1)^2=1$

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$x(x-1)=x^2-x=x^2-x+\dfrac14-\dfrac14=\left(x-\dfrac12\right)^2-\dfrac14$

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The second equation gives no real-valued solutions because squaring any real number gives a positive real number. (I'm assuming we don't care about complex solutions.) So we're left with only

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Then when $x=\tan\theta$ , we can find $\cot\theta$ by taking the reciprocal, so we get

$\boxed{\cot\theta=\dfrac2{1+\sqrt5}\text{ or }\cot\theta=\dfrac2{1-\sqrt5}}$

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