Solve the equation \( \sin(2x)+5=5.83 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
  \(\sin\left(2x\right)+5=5.83\)
- step1: Move the constant to the right side:
  \(\sin\left(2x\right)=5.83-5\)
- step2: Subtract the numbers:
  \(\sin\left(2x\right)=0.83\)
- step3: Use the inverse trigonometric function:
  \(2x=\arcsin\left(0.83\right)\)
- step4: Calculate:
  \(\begin{align}&2x=\arcsin\left(0.83\right)\\&2x=-\arcsin\left(0.83\right)+\pi \end{align}\)
- step5: Add the period:
  \(\begin{align}&2x=\arcsin\left(0.83\right)+2k\pi ,k \in \mathbb{Z}\\&2x=-\arcsin\left(0.83\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\)
- step6: Calculate:
  \(\begin{align}&x=\frac{\arcsin\left(0.83\right)}{2}+k\pi ,k \in \mathbb{Z}\\&2x=-\arcsin\left(0.83\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\)
- step7: Calculate:
  \(\begin{align}&x=\frac{\arcsin\left(0.83\right)}{2}+k\pi ,k \in \mathbb{Z}\\&x=\frac{-\arcsin\left(0.83\right)+\pi }{2}+k\pi ,k \in \mathbb{Z}\end{align}\)
- step8: Find the union:
  \(x=\left\{ \begin{array}{l}\frac{\arcsin\left(0.83\right)}{2}+k\pi \\\frac{-\arcsin\left(0.83\right)+\pi }{2}+k\pi \end{array}\right.,k \in \mathbb{Z}\)
- step9: Simplify:
  \(x\approx \left\{ \begin{array}{l}0.489554+k\pi \\1.081242+k\pi \end{array}\right.,k \in \mathbb{Z}\)
The solution to the equation \( \sin(2x) + 5 = 5.83 \) is \( x \approx 0.489554 + k\pi \) or \( x \approx 1.081242 + k\pi \), where \( k \) is an integer.

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Question

7. \( \sin 2 x+5=5,83 \)

Ask by Collins Pritchard. in South Africa

Feb 20,2025

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Tutor-Verified Answer

Answer

The solutions are \( x \approx 0.489554 + k\pi \) or \( x \approx 1.081242 + k\pi \), where \( k \) is any whole number.

Solution

Solve the equation \( \sin(2x)+5=5.83 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sin\left(2x\right)+5=5.83\) - step1: Move the constant to the right side: \(\sin\left(2x\right)=5.83-5\) - step2: Subtract the numbers: \(\sin\left(2x\right)=0.83\) - step3: Use the inverse trigonometric function: \(2x=\arcsin\left(0.83\right)\) - step4: Calculate: \(\begin{align}&2x=\arcsin\left(0.83\right)\\&2x=-\arcsin\left(0.83\right)+\pi \end{align}\) - step5: Add the period: \(\begin{align}&2x=\arcsin\left(0.83\right)+2k\pi ,k \in \mathbb{Z}\\&2x=-\arcsin\left(0.83\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step6: Calculate: \(\begin{align}&x=\frac{\arcsin\left(0.83\right)}{2}+k\pi ,k \in \mathbb{Z}\\&2x=-\arcsin\left(0.83\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step7: Calculate: \(\begin{align}&x=\frac{\arcsin\left(0.83\right)}{2}+k\pi ,k \in \mathbb{Z}\\&x=\frac{-\arcsin\left(0.83\right)+\pi }{2}+k\pi ,k \in \mathbb{Z}\end{align}\) - step8: Find the union: \(x=\left\{ \begin{array}{l}\frac{\arcsin\left(0.83\right)}{2}+k\pi \\\frac{-\arcsin\left(0.83\right)+\pi }{2}+k\pi \end{array}\right.,k \in \mathbb{Z}\) - step9: Simplify: \(x\approx \left\{ \begin{array}{l}0.489554+k\pi \\1.081242+k\pi \end{array}\right.,k \in \mathbb{Z}\) The solution to the equation \( \sin(2x) + 5 = 5.83 \) is \( x \approx 0.489554 + k\pi \) or \( x \approx 1.081242 + k\pi \), where \( k \) is an integer.

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The Deep Dive

To solve the equation \( \sin 2x + 5 = 5.83 \), we first isolate the sine function. Subtract 5 from both sides: \(\sin 2x = 5.83 - 5 = 0.83\). Now, we find the angle \(2x\) whose sine is 0.83. Using the inverse sine function, we have: \(2x = \arcsin(0.83)\). Then, we can find the principal value and also consider the periodic nature of the sine function. Thus, \(2x \approx 56.44^\circ\) or in radians \(\approx 0.986\) radians. Don't forget that the sine function is positive in both the first and second quadrants, so also consider: \(2x = 180^\circ - 56.44^\circ = 123.56^\circ\) or in radians \(2x \approx 2.155\) radians. Finally, divide by 2 to find \(x\): \(x \approx 28.22^\circ\) or \(x \approx 61.78^\circ\) (in radians, \(x \approx 0.493\) and \(x \approx 1.089\)). And don’t forget to check the periodic solution by including \(n\) in the sine periodicity formula! Now you've got two angles for \(x\), and the world of trigonometry just keeps spinning!

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