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)=
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).
To find the exact value of sin(θ−ϕ), we will use the trigonometric identities.
We know that tanθ=5/12, so let's find the other trigonometric values for θ. First, let's find the value of sinθ.
Using the identity tanθ = sinθ/cosθ, we can rewrite tanθ as sinθ/cosθ:
5/12 = sinθ/cosθ
Next, let's find the value of cosθ. Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can substitute sinθ = 5/12 into the equation:
(5/12)^2 + cos^2θ = 1
25/144 + cos^2θ = 1
cos^2θ = 1 - 25/144
cos^2θ = 144/144 - 25/144
cos^2θ = 119/144
Taking the square root of both sides, we get:
cosθ = ±√(119/144)
Since θ is in the first quadrant, cosθ will be positive. Therefore:
cosθ = √(119/144)
Similarly, we know cotϕ=11/13, so let's find the other trigonometric values for ϕ. First, let's find the value of cosϕ.
Using the identity cotϕ = cosϕ/sinϕ, we can rewrite cotϕ as cosϕ/sinϕ:
11/13 = cosϕ/sinϕ
Next, let's find the value of sinϕ. Using the Pythagorean identity sin^2ϕ + cos^2ϕ = 1, we can substitute cosϕ = 11/13 into the equation:
sin^2ϕ + (11/13)^2 = 1
sin^2ϕ + 121/169 = 1
sin^2ϕ = 169/169 - 121/169
sin^2ϕ = 48/169
Taking the square root of both sides, we get:
sinϕ = ±√(48/169)
Since ϕ is in the first quadrant, sinϕ will be positive. Therefore:
sinϕ = √(48/169)
Now, let's find the value of sin(θ−ϕ). Using the trigonometric identity sin(θ−ϕ) = sinθcosϕ - cosθsinϕ, we can substitute the known values:
sin(θ−ϕ) = (sinθ)(cosϕ) - (cosθ)(sinϕ)
sin(θ−ϕ) = (√(119/144))(√(48/169)) - (√(119/144))(√(48/169))
sin(θ−ϕ) = (√(119/144 * 48/169)) - (√(119/144 * 48/169))
sin(θ−ϕ) = √(119/144 * 48/169) - √(119/144 * 48/169)
sin(θ−ϕ) = √(119*48)/(144*169)
Hence, the exact value of sin(θ−ϕ) is √(119*48)/(144*169).