If tan theta = 9/5 and cot omega = 9/5
Find the exact value of sin (omega-theta)
sorry that should be theta-omega not omega-theta
Cleverly, you note that since
tanθ = cotω, ω = π/2-θ
Thus, θ-ω = θ-(π/2-θ) = π/2+2θ
sin(θ-ω) = -cos2θ = 2sin^θ-1
tanθ = 9/5, so
sinθ = 9/√106
sin(θ-ω) = 162/106 - 1 = 56/106
or, if you must exercise your sum/difference formulas,
sin(θ-ω) = sinθcosω-cosθsinω
tanθ = 9/5, so
sinθ = 9/√106
cosθ = 5/√106
cotω = 9/5, so
sinω = 5/√106
cosω = 9/√106
sin(θ-ω) = 9/√106 * 9/√106 - 5/√106 * 5/√106 = 56/106
To find the exact value of sin(omega-theta), we first need to calculate the values of sin omega and sin theta.
Given:
tan theta = 9/5
cot omega = 9/5
We can use the reciprocal relationships between trigonometric functions to express tan in terms of cot, and vice versa.
Reciprocal relationships:
tan theta = 1 / cot theta
cot omega = 1 / tan omega
Using these relationships, we can rewrite the given equations as:
1 / cot theta = 9/5
1 / tan omega = 9/5
Now, let's solve for cot theta and tan omega:
cot theta = 5/9
tan omega = 5/9
Now that we have the values of cot theta and tan omega, we can find sin theta and sin omega using the Pythagorean identity, which states:
sin^2 x + cos^2 x = 1
For theta:
cot^2 theta + 1 = csc^2 theta
(5/9)^2 + 1 = csc^2 theta
25/81 + 1 = csc^2 theta
106/81 = csc^2 theta
csc theta = ± √(106/81)
For omega:
tan^2 omega + 1 = sec^2 omega
(5/9)^2 + 1 = sec^2 omega
25/81 + 1 = sec^2 omega
106/81 = sec^2 omega
sec omega = ± √(106/81)
From here, we can use the reciprocal identities to find sin theta and sin omega:
sin theta = 1 / csc theta = 1 / (± √(106/81))
sin omega = 1 / sec omega = 1 / (± √(106/81))
Therefore, the exact value of sin(omega-theta) is sin omega * cos theta - cos omega * sin theta, which can be calculated by substituting the values of sin theta and sin omega into the formula.