If tanθ=5/12 and cotϕ=11/13 , with θ and ϕ in the first quadrant, find the exact value of sin(θ−ϕ). Note: Be sure to enter EXACT values sin(θ−ϕ)=

Answer 1

To find the exact value of sin(θ−ϕ), we first need to find the values of sinθ and sinϕ.

Given that tanθ = 5/12, we can use the Pythagorean identity for the tangent function:
tan²θ + 1 = sec²θ
(5/12)² + 1 = sec²θ
25/144 + 1 = sec²θ
169/144 = sec²θ

Taking the square root of both sides, we get:
secθ = √(169/144)
secθ = 13/12

Since secθ = 1/cosθ, we have:
1/cosθ = 13/12
cosθ = 12/13

Using the Pythagorean identity for the cosine function:
sin²θ + cos²θ = 1
sin²θ + (12/13)² = 1
sin²θ + 144/169 = 1
sin²θ = 1 - 144/169
sin²θ = 25/169
sinθ = ±√(25/169)
sinθ = ±5/13

Since θ is in the first quadrant, we take the positive value:
sinθ = 5/13

Similarly, we can find the value of sinϕ using the given value of cotϕ.
cotϕ = 1/tanϕ
cotϕ = 11/13

Using the Pythagorean identity for the cotangent function:
cot²ϕ + 1 = csc²ϕ
(11/13)² + 1 = csc²ϕ
121/169 + 1 = csc²ϕ
290/169 = csc²ϕ

Taking the square root of both sides, we get:
cscϕ = √(290/169)
cscϕ = √290/13

Since cscϕ = 1/sinϕ, we have:
1/sinϕ = √290/13
sinϕ = 13/√290

To find sin(θ−ϕ), we can use the identity sin(A−B) = sinAcosB − cosAsinB.
sin(θ−ϕ) = sinθcosϕ − cosθsinϕ

Plugging in the values we found:
sin(θ−ϕ) = (5/13 * 11/13) - (12/13 * 13/√290)
sin(θ−ϕ) = (55/169) - (156/√290)
sin(θ−ϕ) = (55 - 156√290)/(169√290)

Therefore, the exact value of sin(θ−ϕ) is (55 - 156√290)/(169√290).