There is an angle 𝜃 such that sin 𝜃 = 3/5 and cos 𝜃 = 4/5. In a triangle assume that angle 𝛼 is opposite side a, 𝛽 is opposite b, and 𝛾 is opposite c. Assume that 𝛼 = 𝜃, that b = 3, and that c = 5. Find the length of side a.
using the law of cosines,
a^2 = b^2 + c^2 - 2bc cos𝜃
so plug in your numbers
How do I do that if I don't know what 𝜃 equals?
To find the length of side a, we can use the Law of Sines. The Law of Sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
In this case, we have the angle θ (or α) and we know that sin θ = 3/5. Let's express this in the equation:
a/sin α = c/sin γ
Since α = θ, we can substitute θ for α:
a/sin θ = c/sin γ
We also know that c = 5, so we have:
a/sin θ = 5/sin γ
Now, we need to find sin γ. Since the sum of the angles in a triangle is 180 degrees, we know that γ = 180 - α - β.
We know α = θ, so γ = 180 - θ - β.
To find β, we can use the fact that cos θ = 4/5. The cosine function is positive in the first and fourth quadrants, so θ is in the first quadrant. In the first quadrant, sin θ is positive, so sin θ = sqrt(1 - cos^2 θ) = sqrt(1 - (4/5)^2) = sqrt(1 - 16/25) = sqrt(9/25) = 3/5.
Now we can find β:
cos β = cos (180 - θ - β)
= -cos (θ + β)
= -[cos θ cos β - sin θ sin β]
= -(4/5)cos β - (3/5)sin β
Simplifying:
cos β + (4/5)cos β = (3/5)sin β
(9/5)cos β = (3/5)sin β
9cos β = sin β
Dividing both sides by cos β:
9 = tan β
Taking the inverse tangent of both sides:
β = arctan(9)
Now, we can substitute γ = 180 - θ - β and β = arctan(9) into our equation:
a/sin θ = 5/sin γ
To find sin θ and sin γ, we can use the Pythagorean identity:
sin^2 θ + cos^2 θ = 1
Since sin θ = 3/5, we can solve for cos θ:
(3/5)^2 + cos^2 θ = 1
9/25 + cos^2 θ = 1
cos^2 θ = 1 - 9/25
cos^2 θ = 16/25
cos θ = 4/5 (positive in the first quadrant)
Now we can find sin γ:
sin γ = sin (180 - θ - β)
= sin (arcsin(3/5) + arctan(9))
= sin (θ + β)
= sin θ cos β + cos θ sin β
= (3/5)(-4/5) + (4/5)(9/41)
= -(12/25) + (36/25)
= 24/25
Substituting all the known values into our equation:
a/(3/5) = 5/(24/25)
To solve for a, we need to simplify:
(a/(3/5)) x (25/24) = 5
Multiplying both sides by (3/5) and (24/25):
a = 3 x 5 x 24
a = 360/5
a = 72
Therefore, the length of side a is 72.
really? Who said you have to know 𝜃?
Did you read what you posted? cos𝜃 = 4/5
so use that!