Use the half angle formula to find the exact value of
tan 7 pie/8
How do you tidy it up?
To find the exact value of tan (7π/8) using the half-angle formula, you need to identify the appropriate half-angle formula for tangent.
The half-angle formula for tangent is:
tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ))
In this case, θ = 7π/8.
Step 1: Calculate the value of cos(θ)
cos(7π/8) = cos(π - 7π/8)
Since cos is an even function, cos(π - 7π/8) = cos(7π/8)
= cos(8π/8 - 7π/8)
= cos(π/8)
Step 2: Substitute the value cos(θ) into the half-angle formula
tan(7π/16) = ±√((1 - cos(π/8)) / (1 + cos(π/8)))
= ±√((1 - cos(π/8)) / (1 + cos(π/8)))
Step 3: Simplify the expression
To simplify the expression, we need to rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator:
= ±√((1 - cos(π/8)) / (1 + cos(π/8))) * (√(1 + cos(π/8)) / √(1 + cos(π/8)))
= ±√((1 - cos^2(π/8)) / (1 - cos^2(π/8)))
= ±√(sin^2(π/8) / sin^2(π/8))
= ±√1
= ±1
Therefore, the exact value of tan(7π/8) using the half-angle formula is ±1.
To find the exact value of tan(7π/8) using the half-angle formula, we can first start by considering the half-angle formula for tangent. The formula states that for an angle θ/2, we have:
tan(θ/2) = ± √((1 - cosθ) / (1 + cosθ))
Now, let's apply this formula to find tan(7π/8):
First, we need to determine θ/2. In this case, θ/2 = (7π/8) / 2 = 7π/16.
Next, we need to find the value of cos(7π/8). To do this, we can use the sum-to-product formula for cosine:
cos(2θ) = 2cos²θ - 1
In this case, we can use the formula for cos(θ):
cos(2θ) = 2cos²(θ/2) - 1
Therefore:
cos(7π/8) = cos(2 * (7π/16))
= 2cos²(7π/16) - 1
To find cos(7π/16), we can use the half-angle formula for cosine:
cos(θ/2) = ± √((1 + cosθ) / 2)
In this case, θ = 7π/8, so:
cos(7π/16) = ± √((1 + cos(7π/8)) / 2)
Plug this value back into the formula for cos(7π/8) to solve for cos(7π/8). Then substitute the value of cos(7π/8) into the formula for tan(7π/8) to find the exact value of tan(7π/8).
Since these calculations involve multiple steps, it might be more effective to use a calculator or an online tool to compute the exact value of tan(7π/8).
tan(7pi/8) = tan(7pi/4/2)
hence
tan(7pi/8)=sin(7pi/4)/(1+cos(7pi/4))
sin(7pi/4)=-(root2)/2
cos(7pi/4)=(root2)/2
so
tan(7pi/8)= -(root2)/2/(1+(root2)/2)
and then tidy up.
Sorry about the expressions, not easy to do in text.