Find all solutions of the equation in the interval?

Question

1)6 sin^2 x + 9 sin x + 3 = 0

2)7sinx= cot(−x)

3)4secx + 4tanx = 4

Answer 1

6 sin² x + 9 sin x + 3 = 0

Let sin² x = y

6y² + 9y + 3 = 0

Divide each side by 3.

2y² + 3y + 1 = 0

Now solve the factor method.

2y² + 2y + y + 1 = 0

2y(y + 1) +1(y + 1) = 0

Take out common factors.

(2y + 1)(y + 1) = 0

2y +1 = 0 or y +1 = 0

2y +1 = 0

Subtract 1 from each side.

2y = -1

Divide each side by 2.

y = -1/2

And y + 1 = 0

Subtract 1 from each side.

y = -1

There fore y = -1/2 and y = -1

But y = sin x

So, sin x = -1/2 and sin x = -1

Trigonometric table in sin(11π/6) = -1/2 and sin(3π/2) = -1

There fore sin x = sin(11π/6) and sin x = sin(3π/2)

Cancel common terms.

x = 11π/6 and x = 3π/2

The equation in the interval is [3π/2, 11π/6]

Comments

Let sin x = y

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sin(x) = - 1/2 or sin(x) = - 1.

x = sin^{- 1}(- 1/2) or x = sin^{- 1}(- 1).

The function sin(x) has a period of 2π, first find all solutions in the interval (0, 2π).

The function sin(x) is negative in third and fourth quadrant.

• sin(x) = - 1/2.

In third Quadrant, π ≤ x ≤ 3π/2.

- 1/2 = - sin (π/6) = sin (π + π/6) = sin (7π/6).

In fourth Quadrant, 3π/2 ≤ x ≤ 2π.

- 1/2 = - sin (π/6) = sin (2π - π/6) = sin (11π/6).

So, the general solutions are x = 2nπ + 7π/6, x = 2nπ + 11π/6 where n ∈ Z.

• sin(x) = - 1.

In third Quadrant, π ≤ x ≤ 3π/2.

- 1 = - sin (π/2) = sin (π + π/2) = sin (3π/2).

In fourth Quadrant, 3π/2 ≤ x ≤ 2π.

- 1 = - sin (π/2) = sin (2π - π/2) = sin (3π/2).

So, the general solution is x = 2nπ + 3π/2, where n ∈ Z.

The general solutions of 6 sin² x + 9 sin x + 3 = 0 are x = 2nπ + 7π/6, x = 2nπ + 11π/6 and x = 2nπ + 3π/2, where n ∈ Z.

Answer 2

7sinx= cot(−x)

Odd/Even Identities: cot (–x) = –cot x

7sinx= - cot(x)

Ratio Identities: cot(x) = cos(x) / sin(x)

7sinx= - cos(x) / sin(x)

Multiply each side by sin(x).

7sin²x = - cos(x)

Add cos(x) to each side.

7sin²x + cos(x) = 0

Pythagorean Identities: sin² θ + cos² θ = 1⇒ sin²x = 1 - cos²x

7(1 - cos²x) + cos(x) = 0

7 - 7cos²x + cos(x) = 0

Multiply each side by negative one.

7cos²x - cos(x) - 7 = 0

Let cos x = y

7y² - y - 7 = 0

Now solve the factor method.

Compare equation with standard from ax²+bx+c=0 and write the coefficients.

a = 7, b= -1 and c = -7.

The quadratic formula: x = [-b + √(b² - 4ac)] / 2a

y = [-(-1)+ √((-1)² - 4(7)(-7))] / 2(7)]

y = [1 + √(1 + 196)] / 14]

y = [1 + √(197)] / 14]

y = [1 + √(197)] / 14] or y = [1 - √(197)] / 14]

Simplify

y = 1.0739 (the nearest value is 1.0739=1) or y = 0.9311

But y = cos x

cos(x) = 1 or cos(x) = 0.9311

Trigonometric table of values cos 0= 0 and cos21 = 0.9311

cos x = cos 0 ⇒x = 0

cos x = cos21 ⇒x = 21

The equation in the interval is [0, 21]

Comments

cos 90 = 0

Answer 3

4secx + 4tanx = 4

Take out common term 4.

4(secx + tanx) = 4

Divide each side by 4.

secx + tanx = 1

Reciprocal Identities: secθ = 1/cosθ and Ratio Identities: tanθ = sinθ/cosθ

1/cos(x) + sin(x)/cos(x) = 1

Remrite the expression with common denominator

(1 + sin(x)) / cos(x) = 1

Multiply each side by cos(x).

1 + sin(x) = cos(x)

.Subtract 'sin(x)' from each side.

1 = cos(x) - sin(x)

Apply square each side.

1² = [cos(x) - sin(x)]²

cos²x + sin²x - 2cos(x)sin(x )= 1

Pythagorean Identities: sin² θ + cos² θ = 1

1 - 2sin(x)cos(x) = 1

.Subtract 1 from each side.

-2sin(x)cos(x) = 0

Divide each side by -2.

sin(x)cos(x) = 0

sin(x) = 0 or cos(x) = 0

Trigonometric table of values sin0 = 0 and cos 90= 0

sin(x) = sin(0) ⇒ x = 0

cos(x) = cos(90) ⇒ x = 90

The equation in the interval is [0, 90]

Comments

good work!!!

Answer 4

• 2).

The trigonometric equation is .

Let, , then .

.

Answer 5

Contd...........

Let, ,

.

, is a quadratic, use quadratic equation to find roots of the related quadratic function.

The solution of the quadratic equation : is .

Compare the equation with .

.

Put  .

put  .

.

.

The general solution is , where, .

Answer 6

(3).

Solve for x:

4 sec(x)+4 tan(x) = 4

Move everything to the left hand side.

Subtract 4 from both sides:

-4+4 sec(x)+4 tan(x) = 0

Factor the left hand side.

Factor constant terms from the left hand side:

4 (-1+sec(x)+tan(x)) = 0

Divide both sides by a constant to simplify the equation.

Divide both sides by 4:

-1+sec(x)+tan(x) = 0

Transform -1+sec(x)+tan(x) = 0 into a rational equation via the Weierstrass substitution.

Substitute y = tan(x/2). Then sin(x) = (2 y)/(y^2+1) and cos(x) = (1-y^2)/(y^2+1):

-(2 y)/(y-1) = 0

Divide both sides by a constant to simplify the equation.

Divide both sides by -2:

y/(y-1) = 0

Multiply both sides by a polynomial to clear fractions.

Multiply both sides by y-1:

y = 0

Perform back substitution on y = 0.

Substitute back for y = tan(x/2):

tan(x/2) = 0

Eliminate the tangent from the left hand side.

Take the inverse tangent of both sides:

x/2 = pi n  for  n element Z

Solve for x.

Multiply both sides by 2:

Answer: x = 2 pi n  for  n element Z

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