Find sin B

[asy] size(200); pair A,B,C,D; A=(0,8); B=(-15,0); C=(6,0); D=origin; draw(A--B--C--A--D); draw(rightanglemark(C,D,A,20)); dot("$A$",A,N); dot("$B$",B,W); dot("$C$",C,E); dot("$D$",D,S); label("15",B--D,S); label("$6$",D--C,S); label("10",A--C,NE); [/asy]

 Jun 13, 2024

Right Triangle Identification: We are given that D is the foot of the altitude from A to BC. This implies that ∠ADC=90∘. Therefore, triangle ADC is a right triangle.


Soh Cah Toa: We are asked to find sin B, which is the opposite side (AC) divided by the hypotenuse (in this case, we don't know the length of BC, but we can find it).


Pythagorean Theorem in Right Triangle ADC: We know AC = 10 and CD = 6. Since this is a right triangle, we can use the Pythagorean Theorem to find the length of the missing side (AD):


AD^2 + CD^2 = AC^2


Substitute the known values: AD^2 + 6^2 = 10^2


Solve for AD: AD^2 = 100 - 36 = 64 AD = 64​=8


Pythagorean Theorem to Find Hypotenuse (BC): Now that we know the length of a leg (AD) in right triangle ABC, we can find the length of the hypotenuse (BC) using the Pythagorean Theorem again:


BC^2 = AB^2 + AC^2 (We don't know the length of AB, but we can find BC)


Substitute the known values: BC^2 = 8^2 + 10^2


Solve for BC: BC^2 = 64 + 100 = 164 BC = 164​ (We can leave it in this form for now)


Finding sin B: Now that we know both AC and BC, we can find sin B:


sin B = AC / BC


Substitute the known values: sin B = 10 / sqrt(164​)


Since we cannot simplify 164​ further, the answer is sin B = 10/sqrt(164).

 Jun 13, 2024

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