Solve 7 sin (2a) + 10 cos (a) = 0 for all solutions 0 ≤ a < 2π. Give your answers separted by commas. Round answers to at least two decimal places. a=?

Question

Would someone be able to explain this process? Thank you so much!

Answer 1

To solve the equation 7 sin(2a) + 10 cos(a) = 0 for 0 ≤ a < 2π, follow these steps:

1. Use the double-angle identity for sine:

The double-angle identity is sin(2a) = 2 sin(a) cos(a). Substitute this into the equation:

7 sin(2a) + 10 cos(a) = 0

7 * 2 sin(a) cos(a) + 10 cos(a) = 0

14 sin(a) cos(a) + 10 cos(a) = 0

2. Factor out cos(a):

cos(a) (14 sin(a) + 10) = 0

3. Solve for cos(a) = 0:

cos(a) = 0

The solutions for cos(a) = 0 are:

a = π/2

a = 3π/2

4. Solve for 14 sin(a) + 10 = 0:

14 sin(a) + 10 = 0

14 sin(a) = -10

sin(a) = -10/14 = -5/7

To find a when sin(a) = -5/7, use the inverse sine function:

θ = arcsin(-5/7) ≈ -0.756 radians

Since sin(a) is negative, 'a' will be in the third and fourth quadrants. Compute:

In the third quadrant: a = π + θ ≈ π - 0.756 ≈ 2.385 radians

In the fourth quadrant: a = 2π - θ ≈ 2π + 0.756 ≈ 5.527 radians

5. Compile all solutions:

a ≈ π/2 ≈ 1.57 radians

a ≈ 3π/2 ≈ 4.71 radians

a ≈ 2.385 radians

a ≈ 5.527 radians

Thus, the solutions for 0 ≤ a < 2π, rounded to two decimal places, are approximately:

1.57, 4.71, 2.39, 5.53

Answer 2

7sin2a + 10cosa = 0

7(2sinacosa) + 10cosa = 0

cosA(14sina +10) = 0

cosA = 0, a = 90 or 270 degrees

14sina +10=0, sina= -10/14 = -5/7, a = -45.59 or 360-45.59= 314.41 or 180-45.59 = 134.41

or

graph 7sinx +10cosx = 0, x intercept = about 2.35 radians = a

1 radian = 180/pi degrees= about 57.3 degrees

2.35 radians = 2.35(180)/pi = about 134.41 degrees

Answer 3

Use the identity sin(2a) = 2sinacosa.

So, 14sinacosa + 10cosa = 0.

cosa(14sina + 10) = 0

cosa = 0 or sina = -5/7

cosa = 0 when a = π/2 or 3π/2

sina = -5/7 when a = π + Sin^-1(5/7) or 2π - Sin^-1(5/7)