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Sagot :
By definition of tangent,
tan(θ - x) = sin(θ - x) / cos(θ - x)
Expand the sine and cosine terms using the angle sum identities,
sin(x ± y) = sin(x) cos(y) ± cos(x) sin(y)
cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y)
from which we get
tan(θ - x) = (sin(θ) cos(x) - cos(θ) sin(x)) / (cos(θ) cos(x) + sin(θ) sin(x))
Also recall the Pythagorean identity,
cos²(x) + sin²(x) = 1
from which we have two possible values for each of cos(θ) and sin(x):
cos(θ) = ± √(1 - sin²(θ)) = ± 3/5
sin(x) = ± √(1 - cos²(x)) = ± 12/13
Since there are 2 choices each for cos(θ) and sin(x), we'll have 4 possible values of tan(θ - x) :
• cos(θ) = 3/5, sin(x) = 12/13 :
tan(θ - x) = -56/33
• cos(θ) = -3/5, sin(x) = 12/13 :
tan(θ - x) = 16/63
• cos(θ) = 3/5, sin(x) = -12/13 :
tan(θ - x) = -16/63
• cos(θ) = -3/5, sin(x) = -12/13 :
tan(θ - x) = 56/33
- cosø=3/5
- sinx=12/13
Now
cos(ø-x)
- cosøcosx+sinøsinx
- (3/5)(-5/13)+(4/5)(12/13)
- (33/65)
sin(ø-x)
- sinøsinx-cosøcosx
- 48/65+33/65
- 81/65
So
tan(ø-x)
- sin(ø-x)/cos(ø-x)
- 81/65÷33/65
- 81/33
- 27/11
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