Answer:
-sqrt((1-(sqrt(2)/2)/1+(sqrt(2)/2))
Step-by-step explanation:
tan (7pi/8) = tan (7pi/4/2)
Which equals...
-sqrt((1-cos((7pi/4))/1+cos((7pi/4))
And just simplify into...
-sqrt((1-(sqrt(2)/2)/1+(sqrt(2)/2))
Answer:
The exact form of [tex]\tan(\frac{7\pi}{8})[/tex] is [tex]-\sqrt{2}+1[/tex]
Step-by-step explanation:
We need to find the exact value of [tex]\tan(\frac{7\pi}{8})[/tex] using half angle identity.
Since, [tex]\frac{7\pi}{8}[/tex] is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.
[tex]\frac{7\pi}{8}[/tex] is not an exact angle.
First, rewrite the angle as the product of [tex]\frac{1}{2}[/tex] and an angle where the values of the six trigonometric functions are known. In this case,
[tex]\frac{7\pi}{8}[/tex] can be written as ;
[tex](\frac{1}{2})\frac{7\pi}{4}[/tex]
Use the half-angle identity for tangent to simplify the expression. The formula states that [tex] \tan \frac{\theta}{2}=\frac{\sin \theta}{1+ \cos \theta}[/tex]
[tex]=\frac{\sin(\frac{7\pi}{4})}{1+ \cos (\frac{7\pi}{4})}[/tex]
Simplify the numerator.
[tex]=\frac{\frac{-\sqrt{2}}{2}}{1+ \cos (\frac{7\pi}{4})}[/tex]
Simplify the denominator.
[tex]=\frac{\frac{-\sqrt{2}}{2}}{\frac{2+\sqrt{2}}{2}}[/tex]
Multiply the numerator by the reciprocal of the denominator.
[tex]\frac{-\sqrt{2}}{2}\times \frac{2}{2+\sqrt{2}}[/tex]
cancel the common factor of 2.
[tex]\frac{-\sqrt{2}}{1}\times \frac{1}{2+\sqrt{2}}[/tex]
Simplify,
[tex]\frac{-\sqrt{2}(2-\sqrt{2})}{2}[/tex]
[tex]\frac{-(2\sqrt{2}-\sqrt{2}\sqrt{2})}{2}[/tex]
[tex]\frac{-(2\sqrt{2}-2)}{2}[/tex]
simplify terms,
[tex]-\sqrt{2}+1[/tex]
Therefore, the exact form of [tex]\tan(\frac{7\pi}{8})[/tex] is [tex]-\sqrt{2}+1[/tex]
