Question

find the acute angle x, given that cos x° =sin 2x°

Answer 1

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sin2x+cos2x=1(1)

The trigonometric formulas for sums of angles:

sin(u+v)=sinucosv+cosusinv(2.1)

cos(u+v)=cosucosv−sinusinv(2.2)

x is acute, meaning that we are in the first quadrant:

sinx>0(3.1)

cosx>0(3.2)

The value of cos2x is known:

cos2x=119169(4)

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Now, let’s try to deduce a formula for sinx as a function of cos2x from this.

We see that we can get an expression containing cos2x from (2.2) , by setting u=v=x . This gives:

cos(x+x)=cosxcosx−sinxsinx(2.2-1)

Simplifying this:

cos2x=cos2x−sin2x(2.2-2)

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The only unwanted thing in this equation now would be cos2x . Luckily, we can get rid of this using (1) . First we have to manipulate it by moving the sine-part to the right side:

cos2x=1−sin2x(1-1)

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Now, we can plug this in to (2.2−2) :

cos2x=(1−sin2x)−sin2x(5-1)

Simplifying this:

cos2x=1−2sin2x(5-2)

Solving for sinx :

sinx=±1−cos2x2−−−−−−√(5-3)

Now, using (3.1) , this simplifies to only one solution:

sinx=1−cos2x2−−−−−−√(5-4)

———

At last, we can plug (4) into (5–4) :

sinx=1−1191692−−−−−√=25169−−−√=513

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