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<span>11. Find the exact value by using a half-angle identity.
</span><span>sin (22.5)
</span><span>the sine half-angle formula
</span>
12. Find all solutions to the equation in the interval [0, 2π)
</span>cos x = sin 2x
cosx-sin 2x=0
<span>using a graphical tool
</span>in the interval [0, 2π)
</span></span>
</span></span>
</span><span>sin (22.5)
</span><span>the sine half-angle formula
</span>
sin<span>(x/2)</span>=±((1−cos(x))/2) ^0.5 cos 45=(2^0.5)/2
sin(22.5)=±((1−cos(45))/2) ^0.5
sin(22.5)=±((2-2^0.5))^0.5/2
sin(22.5)=±0.3826834324
<span>12. Find all solutions to the equation in the interval [0, 2π)
</span>cos x = sin 2x
cosx-sin 2x=0
<span>using a graphical tool
</span>in the interval [0, 2π)
<span>the solutions are
x1=0----------------not
solution
x2=</span>π/6------------ not solution<span>
x3=</span>π/2------------ is a solution<span>
x4=5</span>π/6---------- not solution<span>
x5=3</span>π/2---------- is a solution
</span></span>
13. Rewrite with only sin x and cos x. sin(2x) = 2*sin(x)*cos(x)
sin 2x - cos x=2*sin(x)*cos(x)- cos x= cos x*(2*sin(x)-1)
<span>the answer is the letter <span>c) cos x (2 sin x - 1)</span></span>
<span>14. Verify the
identity.
cosine of x divided by quantity one plus sine of x plus quantity one plus sine
of x divided by cosine of x equals two times secant of x</span>.
cosx/(1+sinx) + (1+sinx)/cosx
<span>
= (cosx * cosx + (1+sinx)(1+sinx)) / (cosx (1+sinx))
= (cos²x + sin²x + 2 sinx + 1) / (cosx (1+sinx))
= (1 + 2 sinx + 1) / (cosx (1+sinx))
= (2 + 2 sinx) / (cosx (1+sinx))
= 2 (1+sinx) / (cosx (1+sinx))
= 2/cosx
<span>= 2 secx Ok is correct</span></span>