Answer:
sin(θ-φ) = -(21/1105)√85
Step-by-step explanation:
There are a few relations among trig functions that are useful here.
- tan = sin/cos
- cot = cos/sin
- tan²+1 = 1/cos²
- cot²+1 = 1/sin²
- sin(α-β) = sin(α)cos(β) - cos(α)sin(β)
Using the above relations, we observe that ...
tan(θ)cot(φ) = sin(θ)cos(φ)/(cos(θ)sin(φ))
Subtracting 1 gives ...
tan(θ)cot(φ) -1 = (sin(θ)cos(φ) -cos(θ)sin(φ))/(cos(θ)sin(φ))
= sin(θ-φ)/(cos(θ)sin(φ))
We can find the values of the denominator terms:
cos(θ) = √(1/(tan²(θ)+1)) = √(1/((12/5)² +1)) = √(25/169) = 5/13
sin(φ) = √(1/(cot²(φ)+1)) = √(1/((2/9)² +1)) = √(81/85) = (9√85)/85
Then the desired sine value is ...
sin(θ-φ) = (tan(θ)cot(φ) -1)(cos(θ)sin(φ)) = ((12/5)(2/9) -1)(5/13)(9√85/85)
= -21√85/(13·85)
sin(θ-φ) = -(21/1105)√85
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Check
θ = arctan(12/5) ≈ 67.3801°
φ = arccot(2/9) ≈ 77.4712°
θ -φ ≈ -10.0911°
sin(θ -φ) ≈ sin(-10.0911°) ≈ -0.175213
-(21/1105)√85 ≈ -0.175213
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Additional comment
Perhaps more straightforward would be the computation of the sine and cosine of each angle, then find the difference of products according to the sine formula for the difference of angles. This takes more explanation, but doesn't require as much working of individual trig functions. The straightforward approach would give ...
sin(θ -φ) = (12/13)(2/√85) -(5/13)(9/√85) = -21/(13√85) . . . same as above
You'd have to develop the other two trig functions of θ and φ to get this.
