# Given that #tan2A = 3/4# and that the angle #A# is acute, calculate, without using tables, the values of: (i) #cos2A# (ii) #sinA# (iii) #tanA# (iv) #tan3A# ?

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Given that ( \tan(2A) = \frac{3}{4} ), and angle ( A ) is acute, we can use trigonometric identities and relationships to find the values requested.

(i) To find ( \cos(2A) ), we can use the identity ( \cos(2A) = 1 - 2\sin^2(A) ). First, we need to find ( \sin(A) ) using the fact that ( \tan(2A) = \frac{3}{4} ). Since ( \tan(2A) = \frac{2\tan(A)}{1 - \tan^2(A)} ), we can solve for ( \tan(A) ) and then find ( \sin(A) ) using ( \sin(A) = \frac{\tan(A)}{\sqrt{1 + \tan^2(A)}} ).

(ii) Once we have ( \sin(A) ), we can find ( \sin(2A) ) using ( \sin(2A) = 2\sin(A)\cos(A) ). Then ( \sin(2A) ) can be used to find ( \cos(2A) ) using ( \cos(2A) = \sqrt{1 - \sin^2(2A)} ).

(iii) ( \tan(A) ) is straightforward to find once ( \sin(A) ) and ( \cos(A) ) are known.

(iv) ( \tan(3A) ) can be expressed in terms of ( \tan(A) ) using the tangent addition formula: ( \tan(3A) = \frac{\tan(A) + \tan(2A)}{1 - \tan(A)\tan(2A)} ).

We can now use these steps to find the requested values.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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