Find the Angle Whose Sine is the Product of the Sine and Cosine of Two Other Angles
Find the Angle Whose Sine is the Product of the Sine and Cosine of Two Other Angles
This Demonstration shows a geometric solution of the trigonometric equation , where is unknown.
sinγ=sinμcosδ
γ
Draw a right-angle triangle with the hypotenuse and . Then . Let be the circumcircle of .
ABC
AC=1
∠ACB=μ
AB=sinμ
σ
1
ABC
Transform the equation to and interpret it as the law of sines for a triangle with of length 1 opposite the angle and side of length opposite the unknown angle .
sinγ=sinμsin(π/2-δ)
ACH
AC
π/2-δ
AH
sinμ
γ
To find the point , draw a circle with inscribed angle over the chord . Let be the center of . Construct the point on so that . By the law of sines, the angle at is a solution for .
H
σ
2
π/2-δ
AC
S
σ
2
H
σ
2
AH=AB=sinμ
∠HCA
C
γ
If is the intersection of and , then . You can choose to show the point .
G
σ
1
CH
AG=sinγ=sinμcosδ
G