Q.301. Simultaneous equations with 3 unknowns and trigo functions

Question 301.
V1f Cos37 + V2f Cos φ = 3.5 x 10^5
V1f Sin37 - V2f Sinφ =0
V1f^2 + V2f^2 = (3.5 x 10^5) ^2.

Answer 301.
For better visibility, let V1f = x and V2f = y and 3.5 x 10^5 = a
=> x cos37 + y cosφ = a
=> y cosφ = a - x cos37 ... (1)

x sin37 - y sinφ = 0
=> y sinφ = x sin37 ... (2) and

x^2 + y^2 = a^2 ... (3)

Squarring and adding eqns. (1) and (2),
y^2 = x^2 + a^2 - 2ax cos37 ... (4)

Solving eqns. (3) and (4) for x,
a^2 - x^2 = x^2 + a^2 - 2ax cos37
=> 2x^2 - 2axcos37 = 0
=> x = 0 or acos37 = 3.5 x 10^5 cos37 = 2.795 x 10^5
x = 0 => y = ± 3.5 x 10^5 and
x = 2.795 x 10^5 => y = ± 2.107 x 10^5

Answers:
(1) V1f = 0 and V2f = ± 3.5 x 10^5, OR
(2) V1f = 2.795 x 10^5 and V2f = ± 2.107 x 10^5.

Attached link of Wolfram Alpha gives two values of V2f (+ve and -ve) for each value of V1f depending upon the value of angle φ.
Source(s):

Link to YA!