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# ?

Answer 1

#cos2A=4/5#; #sinA=1/sqrt10#; #tanA=1/3# and #tan3A=13/9#

As #tan2A=3/4#, we have
#(2tanA)/(1-tan^2A)=3/4#
or #8tanA=3-3tan^2A#
or #3tan^2A+8tanA-3=0#
or #(3tanA-1)(tanA+3)=0#
i.e. #tanA=1/3# or #tanA=-3#
but as #A# is acute, we cannot have #tanA=-3#
Hence #tanA=1/3#
and #tan3A=tan(A+2A)=(tanA+tan2A)/(1-tanAtan2A)#
= #(1/3+3/4)/(1-1/3xx3/4)=((4+9)/12)/(1-1/4)=13/12xx4/3=13/9#
#cos2A=(cos^2A-sin^2A)/(cos^2A+sin^2A)=(1-tan^2A)/(1+tan^2A)#
= #(1-1/9)/(1+1/9)=(8/9)/(10/9)=8/10=4/5#
As #2cos^2A-1=cos2A# i.e. #cos^2A=(1+4/5)/2=9/10#
#cosA=3/sqrt10#
and #sinA=cosAxxtanA=3/sqrt10xx1/3=1/sqrt10#
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Answer 2

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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Answer from HIX Tutor

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.

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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