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LenaWriter [7]
3 years ago
13

If cot x=v3, calculate (sin x)(cos x)-cos²x

Mathematics
1 answer:
ohaa [14]3 years ago
8 0

Answer:

sin(x)\,cos(x)-cos^2(x)=\frac{\sqrt{3} -3}{4}

Step-by-step explanation:

Recall the Pythagorean identity for cotangent:

1+cot^2(x)=csc^2(x)

where the cosecant is the reciprocal of the sine function. So, since we know the value of the cotangent of the angle, we can derive the value of the square of the sine:

1+cot^2(x)=csc^2(x)\\1+(\sqrt{3})^2=\frac{1}{sin^2(x)}  \\1+3=\frac{1}{sin^2(x)}  \\4=\frac{1}{sin^2(x)}\\sin^2(x)=\frac{1}{4}\\ sin(x)=+/-\frac{1}{2}

We can also use the Pythagorean identity for sin and cos, to find the value of cos^2(x) and of cos(x):

cos^2(x)=1-sin^2(x)\\cos^2(x)=1-\frac{1}{4} \\cos^2(x)=\frac{3}{4}\\cos(x)=+/-\frac{\sqrt{3}}{2}

We also notice that since the cotangent is positive, the angle "x" must be located in either the firs or the third quadrant, where both sine and cosine have the same sign (both positive in the first quadrant, and both negative in the third quadrant.

Then the requested quantity can be written as:

sin(x)\,cos(x)-cos^2(x)=(\frac{1}{2}) \,\frac{\sqrt{3} }{2} -\frac{3}{4} =\frac{\sqrt{3} }{4} -\frac{3}{4}=\frac{\sqrt{3}-3 }{4}

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Answer:

The measures of the angles are 65° and 80°

Step-by-step explanation:

One of the angles is 35°

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With this knowledge we can write an equation.

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Step-by-step explanation:

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Write -2i + (9 - 3i) - (6 - 10i) as a complex number in standard form. 3 - 5i 3 + 5i 3 + 16i 12 - 16i
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Answer:

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