Answer:
Exact Form:
√2−1
Decimal Form:
0.41421356
…
Step-by-step explanation:
Since 9π/8 is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.
9π/8 is not an exact angle
First, rewrite the angle as the product of 1/2 and an angle where the values of the six trigonometric functions are known. In this case, 9π/8 can be rewritten as
(1/2)* 9π/8 tan ((1/2)* 9π/8)
Use the half-angle identity for tangent to simplify the expression. The formula states that
tan (0/2)=sin(0)/1+cos(0) sin(9π/4)/1+cos(9π/4)
Simplify
Remove full rotations of 2π until the angle is between 0 and 2π.
sin(π/4)/1+cos(9π/4)
The exact value of sin(π/4) is √2/2
√2/2/1+cos(9π/4)
Simplify the Denominator
Remove full rotations of 2π until the angle is between 0 and 2π.√2/2/1+cos(π4)
The exact value of cos(π/4) is √2/2.√2/2/1+√2/2
To write 1/1 as a fraction with a common denominator, multiply by 2/2 .√2/2/1/1⋅2/2+√2/2
Write each expression with a common denominator of 2, by multiplying each by an appropriate factor of 1.
Combine.
√2/2/1⋅2/1⋅2+√2/2
Multiply 2 by 1
√2/2/1⋅2/2+√2/2
Combine the numerators over the common denominator.
√2/2/1⋅2+√2/2
Multiply 2 by 1
√2/2/2+√2/2
Multiply the numerator by the reciprocal of the denominator
√2/2 ⋅ 2/2+√2
Cancel the common factor of 2 .
Factor out the greatest common factor 2
√2/2⋅1 ⋅ 2⋅1/2+√2
Cancel the common factor
√2/2⋅1 ⋅ 2⋅1/2+√2
Rewrite the expression.
√2/1 ⋅ 1/2+√2
Simplify
Multiply √2/1 and 1/2+√2
√2/2+√2
Multiply √2/2+√2 by 2−√2/2−√2
Combine
√2(2−√2)/(2+√2)(2−√2)
Expand the denominator using the FOIL method.
√2(2−√2)/4−2√2+√2⋅2−√2^2
Simplify
√2(2−√2)/2
Apply the distributive property
√2⋅2+√2(−√2)/2
Move 2 to the left of the expression √2⋅2.
2⋅√2+√2(−√2)/2
Simplify
√2(−√2) .
Raise √2 to the power of 1 .
2⋅√2−(√2^1√2)/2
Raise √2 to the power of 1 .
2⋅√2−(√2^1√2^1)/2
Use the power rule a^m a^n=a^m+n to combine exponents.
2⋅√2−√2^1+1/2
Add 1 and 1 .
2⋅√2−√2^2/2
Simplify each term.
Multiply 2 by √2 .
2√2−√2^2/2
Rewrite √2^2 as 2 .
2√2−1⋅2/2
Multiply −1 by 2.
2√2−2/2
Reduce the expression by cancelling the common factors.
Factor 2 out of 2√2.
2(√2)−2/2
Factor 2 out of −2.
2(√2)+2⋅−1/2
Factor 2 out of
2(√2)+2(−1)2(√2−1)/2
Cancel the common factors.
Factor 2 out of 2 .
2(√2−1)/2(1)
Cancel the common factor.
2(√2−1)/2⋅1
Rewrite the expression.
√2−1/1
Divide √2−1 by 1 .
√2−1
The result can be shown in multiple forms.
Exact Form:
√2−1
Decimal Form:
0.41421356…
Hope it help. Good luck.