Solve the equation (5)/( x) = 7 − (2 2 x)/( x) for x in the range −90^ x 90^.

Question

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Step 1: Identify the domain and restrictions. The equation involves sin x in the denominator, so sin x ≠ 0. This means x ≠ 0° within the given range of −90° ≤ x ≤ 90°. Step 2: Combine the terms on the left side of the equation. (5)/(sin x) = 7 − (2 cos² x)/(sin x) Add (2 cos² x)/(sin x) to both sides: (5)/(sin x) + (2 cos² x)/(sin x) = 7; (5 + 2 cos² x)/(sin x) = 7; Step 3: Use the trigonometric identity cos² x = 1 − sin² x. Substitute this into the equation: (5 + 2(1 − sin² x))/(sin x) = 7; (5 + 2 − 2 sin² x)/(sin x) = 7; (7 − 2 sin² x)/(sin x) = 7 Step 4: Multiply both sides by sin x. 7 − 2 sin² x = 7 sin x Step 5: Rearrange the equation into a quadratic form in terms of sin x. Move all terms to one side: 2 sin² x + 7 sin x − 7 = 0; Step 6: Solve the quadratic equation for sin x using the quadratic formula y = (−b ± √(b² − 4ac))/(2a), where y = sin x. Here, a=2, b=7, c=−7. sin x = (−7 ± √(7² − 4(2)(−7)))/(2(2)); sin x = (−7 ± √(49 + 56))/(4); sin x = (−7 ± √(105))/(4) Step 7: Calculate the two possible values for sin x. sin x₁ = (−7 + √(105))/(4) ≈ (−7 + 10.247)/(4) ≈ 3.247/4 ≈ 0.81175; sin x₂ = (−7 − √(105))/(4) ≈ (−7 − 10.247)/(4) ≈ (−17.247)/(4) ≈ −4.31175 Step 8: Check for valid values of sin x. The range of sin x is [−1, 1]. sin x₁ ≈ 0.81175 is within the valid range. sin x₂ ≈ −4.31175 is outside the valid range, so this solution is rejected. Step 9: Find the value of x for sin x = 0.81175 within the given range −90° ≤ x ≤ 90°. Since sin x is positive, x must be in the first quadrant. x = arcsin(0.81175) x ≈ 54.26° This value is within the specified range −90° ≤ x ≤ 90° and satisfies the restriction x ≠ 0°. The solution to the equation is x ≈ 54.26°.

Solve the equation (5)/( x) = 7 − (2 2 x)/( x) for x in the range -90^ x 90^.

Answer

Step-by-step solution

Identify the domain and restrictions.

Combine the terms on the left side of the equation.

Use the trigonometric identity $\cos^2 x = 1 - \sin^2 x$ .

Multiply both sides by $\sin x$ .

Rearrange the equation into a quadratic form in terms of $\sin x$ .

Solve the quadratic equation for $\sin x$ using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $y = \sin x$ .

Calculate the two possible values for $\sin x$ .

Check for valid values of $\sin x$ .

Find the value of $x$ for $\sin x = 0.81175$ within the given range $-90^\circ \le x \le 90^\circ$ .

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More Mathematics Questions

Solve the equation (5)/( x) = 7 − (2 2 x)/( x) for x in the range -90^ x 90^.

Answer

Step-by-step solution

Identify the domain and restrictions.

Combine the terms on the left side of the equation.

Use the trigonometric identity cos⁡2x=1−sin⁡2x\cos^2 x = 1 - \sin^2 xcos2x=1−sin2x.

Multiply both sides by sin⁡x\sin xsinx.

Rearrange the equation into a quadratic form in terms of sin⁡x\sin xsinx.

Solve the quadratic equation for sin⁡x\sin xsinx using the quadratic formula y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}y=2a−b±b2−4ac​​, where y=sin⁡xy = \sin xy=sinx.

Calculate the two possible values for sin⁡x\sin xsinx.

Check for valid values of sin⁡x\sin xsinx.

Find the value of xxx for sin⁡x=0.81175\sin x = 0.81175sinx=0.81175 within the given range −90∘≤x≤90∘-90^\circ \le x \le 90^\circ−90∘≤x≤90∘.