What is tan(x - pi/4)?

Question

If sin x= 7/25, and 0 < x < pi/2, what is tan(x - pi/4)?

Answer 1

tan(x - π/4) = (tanx - tanπ/4) / (1 + tanxtanπ/4))

(tanx - 1) / (1 + tanx)

since sinx = 7/25 and x is in quadrant 1, cosx = √[1 - sin²x] = 24/25

So, tanx = sinx / cosx = 7/24

Therefore, tan(x - π/4) = (7/24 - 1) / (1 + 7/24) = (-17/24) / (31/24) = -17/31

Answer 2

Remember that tan(x – y) = (tan x – tan y)/(1 + tan x tan y).

Therefore tan(x – π/4) = (tan x – tan(π/4))/(1 + tan x tan π/4).

And tan(π/4) = 1, so tan(x – π/4) = (tan x – 1)/(1 + tan x).

Since you have sin(x) = 7/25, you have cos(x) = ±√(1 – (7/25)²) = ±24/25, and since x is in the first quadrant, you conclude that cos(x) = 24/25.

Then tan(x) = sin(x) / cos(x) = 7/24.

So, finally, tan(x – π/4) = (tan x – 1)/(1 + tan x) = (7/24 – 1)/(1 + 7/24)

= (7 – 24)/(24 + 7) = –17/31.

(If you're not following the above, one could wonder if it's because your posted question did not specify which aspects of the problem you had difficulty with. Normally you should do that.)

Answer 3

opposite is 7 and hypotenuse is 25.

Then by pythagorean, adjacent is 24.

So then the cosine is 24/25

THen then the angle is T=inverse_sine(7/25) = inverse_cosine(24/25) = 16.2602047083.....

pi/4 = 45 degrees LESS than that, which makes it negative, so then it wraps back around

the unit circle until you get 331.2602047083

the tangent is -0.54838709677446500534565273342049....