Alan

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This is just the equation of a straight line with gradient 7 and y-intercept of -42

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csc is 1/sin so if csc θ = 2 then sin θ = 1/2

cos θ = √(1 - sin²θ) = √(1 - 1/4) =  √(3/4) = (√3)/2   

sec θ = 1/ cos θ

tan θ = sin θ/cos θ =  (1/2)/ (√3)/2 = 1/2*2/√3 = 1/√3

cot θ = 1/tan θ

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3a² + 19a + 30 factors as (3a + 10)(a + 3)

a² + 6a + 9 factors as (a+3)²

The greatest common factor is therefore f(a) = (a + 3), which means that f(a) - a = 3 for all a.

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About what?

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It means the 18s go on forever!  You could, perhaps, use the approximate value 0.192.

Let's call the product of the first 20 positive integers, p.

21 and 22 clearly have common divisors with p (3 and 7 in the case of 21; 2 and 11 in the case of 22).

However, 23 is prime and does not divide p exactly (in fact p/23 has remainder 11), so their greatest common divisor is 1.  So 23 is the smallest positive integer that is greater than 1 and relatively prime to p.

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As 0.19^(1/2) or, perhaps, 0.19^0.5

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I think this might be intended to be 

$$x^2-16x=-55$$

in which case rewrite as:

$$x^2-16x+55=0$$

This factorizes nicely:

$$(x-5)(x-11)=0$$

So we have x = 5 and x = 11 as the two solutions.

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Think of a right-angled triangle in which the adjacent side to theta is √3 and the opposite is 1 (so cot(theta) = √3/1 = √3).

The hypotenuse is then: √(3+1) = √4 = 2.

This means sin(theta) = opposite/hypotenuse = 1/2

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