please help trigonometric-equations.

Question

4)solve

a) tan B=sec B/ csc B

b)sec x- tan x=1-tan x/ cos x

5)Solve 5 sin A - cos A = 0, for 0 ≤ A < 2π (round to the nearest hundredthof a radian).

6) Find the equation of the following graph.

 

Answer 1

4)

b) The trigonometric equation secx - tanx = (1 - tanx)/cosx

(secx - tanx)cosx = 1 - tanx

secx cosx - tanx cosx = 1- tanx

(1/cosx)cosx - tanx cosx = 1 - tanx

1 - tanx cosx = 1 - tanx

1 - 1 = tanx cosx - tanx

tanx cosx - tanx = 0

tanx(cosx - 1) = 0

Apply zero product property.

tanx = 0 and cosx - 1 = 0

case 1 : Solve the equation tanx = 0

The general solution of tan(θ) = 0 is θ = nπ where n is an integer.

In this case θ = x.

x = nπ

Case 2 : Solve the equation cosx - 1 = 0

cosx = 1

cosx = cos(0)

The general solution of cos(θ) = cos(α) is θ = 2nπ± α where n is an integer.

In this case θ = x and α = 0

Solutions are x = 2nπ

Solutions of the equation secx - tanx = (1 - tanx)/cosx are x = nπ and x = 2nπ, wheren is an integer.

Answer 2

4)

a) tanB = secB/cscB

tanB = (1/cosB)/(1/sinB)

tanB = sinB/cosB

tanB = tanB

The above expression is true for all values of B .

Therefore the solutions are all values of B.

Answer 3

5) The trigonometric equation 5 sinA - cosA = 0

5 sinA = cosA

Divide each side by cosA.

5 sinA/cosA = cosA/cosA

5 tanA = 1

tanA = 1/5

tanA = 0.2

A = tan^-1(0.2)

The general solution of tan(θ) = tan(α) is θ = nπ + α.

A = nπ + 0.197

For n = 0, A = (0)π + 0.197

A = 0.197

For n = 1, A = (1)(3.14) + 0.197

A = 3.337

For n = 2, A = (2)(3.14) + 0.197

A = 6.477

The solutions are in the interval 0 ≤ A < 2π are A = 0.2^c and 3.34^c.

Answer 4

6)

Graph of  y = a cos(bx - c) have the following characteristics.

Amplitude |a|, period = 2π/b.

From the graph amplitude of the function a = - 3

Period = 4

y = a cos(bx - c)

Period = 2π/b

2π/b = 4

b = 2π/4

b = π/2

(π/2)x + c = 0

c = - (π/2)x

vertical shift c = 3

The function is y = - 3cos[(π/2)x - 3)].