If sin x= 7/25, and 0 <x< π/2, what is tan(x − π/4)?
I don't get how to do this. We're talking about the "Difference Identity for Tangent". If you could help I'd really appreciate it.
since sinx = 7/25
cosx = 24/25
tanx = 7/24
tan(x - π/4) = (tanx - tan π/4)/(1 + tanx tan(π/4))
= (7/24 - 1)/(1 + 7/24 * 1) = -17/31
makes sense, since x < π/4, x-π/4 is negative
To find the value of tan(x - π/4), we'll need to use the Difference Identity for Tangent, which states:
tan(a - b) = (tan a - tan b) / (1 + tan a * tan b)
In this case, we are given the value of sin(x), so we can use the Pythagorean Identity to find the value of cos(x) and subsequently find the value of tan(x).
Step 1: Find the value of cos(x)
Using the Pythagorean Identity:
cos^2(x) + sin^2(x) = 1
cos^2(x) + (7/25)^2 = 1
cos^2(x) + 49/625 = 1
cos^2(x) = 1 - 49/625
cos^2(x) = 576/625
cos(x) = √(576/625)
cos(x) = 24/25
Step 2: Find the value of tan(x)
Using the formula: tan(x) = sin(x) / cos(x)
tan(x) = (7/25) / (24/25)
tan(x) = 7/24
Step 3: Substitute the values into the Difference Identity for Tangent
tan(x - π/4) = (tan(x) - tan(π/4)) / (1 + tan(x) * tan(π/4))
tan(x - π/4) = (7/24 - 1) / (1 + (7/24) * 1)
Now simplify the expression further:
tan(x - π/4) = (7/24 - 24/24) / (1 + (7/24))
tan(x - π/4) = (-17/24) / (31/24)
Dividing by a fraction is the same as multiplying by its reciprocal:
tan(x - π/4) = (-17/24) * (24/31)
tan(x - π/4) = -17/31
So, tan(x - π/4) equals -17/31.