use law of cosines to show that theta equals inverse cosine of (a^2 + b^2 -484)/(2ab) where a^2=(7+xcos a)^2 + (28-xsin a)^2 and b^2=(7+xcos a)^2 + (xsin a -6)^2
To use the law of cosines to find the angle θ, we need to find the lengths of sides a and b in the given equation.
Given:
a^2 = (7 + xcos(a))^2 + (28 - xsin(a))^2
b^2 = (7 + xcos(a))^2 + (xsin(a) - 6)^2
Let's expand these equations and simplify them before proceeding further.
Expanding a^2:
a^2 = (7 + xcos(a))^2 + (28 - xsin(a))^2
= 49 + 14xcos(a) + x^2cos^2(a) + 784 - 56xsin(a) + x^2sin^2(a)
Expanding b^2:
b^2 = (7 + xcos(a))^2 + (xsin(a) - 6)^2
= 49 + 14xcos(a) + x^2cos^2(a) + 784 - 84xsin(a) + 36 - 12xsin(a) + x^2sin^2(a)
Simplifying both equations, we get:
a^2 = 833 + 14xcos(a) + x^2(cos^2(a) + sin^2(a)) - 56xsin(a)
= 833 + 14xcos(a) + x^2 - 56xsin(a)
b^2 = 869 + 14xcos(a) + x^2(cos^2(a) + sin^2(a)) - 96xsin(a)
= 869 + 14xcos(a) + x^2 - 96xsin(a)
Now, using the Law of Cosines, we have:
c^2 = a^2 + b^2 - 2ab*cos(θ)
Substituting the values we obtained for a^2 and b^2:
c^2 = (833 + 14xcos(a) + x^2 - 56xsin(a)) + (869 + 14xcos(a) + x^2 - 96xsin(a)) - 2 * √((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a))) * cos(θ)
Simplifying further:
c^2 = 1702 + 28xcos(a) + 2x^2 - 152xsin(a) - 2√((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a))) * cos(θ)
Now, we can solve for θ by isolating it in the equation above:
Rearranging the equation:
c^2 - 1702 - 28xcos(a) - 2x^2 + 152xsin(a) = -2√((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a))) * cos(θ)
Dividing both sides by -2√((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a))):
(c^2 - 1702 - 28xcos(a) - 2x^2 + 152xsin(a)) / -2√((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a))) = cos(θ)
Finally, taking the inverse cosine (arccos) of both sides, we get:
θ = arccos((c^2 - 1702 - 28xcos(a) - 2x^2 + 152xsin(a)) / -2√((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a))))
Therefore, θ equals inverse cosine of ((c^2 - 1702 - 28xcos(a) - 2x^2 + 152xsin(a)) / -2√((833 + 14xcos(a) + x^2 - 56xsin(a))(869 + 14xcos(a) + x^2 - 96xsin(a)))).