Question Find the smallest positive solution to the equation $6 \sin (2 x) = 3$ , accurate to at least two decimal places.

Studdy Solution

STEP 1

Assumptions
1. We are solving the equation $6 \sin (2x) = 3$ .
2. We are looking for the smallest positive solution for $x$ .
3. The solution should be accurate to at least two decimal places.

STEP 2

First, we need to isolate $\sin(2x)$ by dividing both sides of the equation by 6.
$\sin(2x) = \frac{3}{6}$

STEP 3

Now, simplify the fraction on the right-hand side.
$\sin(2x) = \frac{1}{2}$

STEP 4

To find $x$ , we need to take the inverse sine (arcsin) of both sides of the equation. However, since we have $2x$ instead of $x$ , we will first find the angle that corresponds to $\sin(\theta) = \frac{1}{2}$ and then divide that angle by 2 to solve for $x$ .
$2x = \arcsin\left(\frac{1}{2}\right)$

STEP 5

The value of $\arcsin\left(\frac{1}{2}\right)$ is known to be $30^\circ$ or $\frac{\pi}{6}$ radians. We will use radians because they are more common in mathematics for trigonometric functions.
$2x = \frac{\pi}{6}$

STEP 6

Now, divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{\pi}{6} \div 2$

STEP 7

Simplify the right-hand side to find the value of $x$ .
$x = \frac{\pi}{12}$

STEP 8

Convert the radian measure to a decimal to get the smallest positive solution for $x$ .
$x \approx \frac{3.14159}{12}$

STEP 9

Perform the division to obtain the decimal value of $x$ .
$x \approx 0.2618$
The smallest positive solution for $x$ accurate to at least two decimal places is approximately $0.26$ radians.

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Question Find the smallest positive solution to the equation $6 \sin (2 x) = 3$ , accurate to at least two decimal places.

Studdy Solution

STEP 1

Assumptions
1. We are solving the equation $6 \sin (2x) = 3$ .
2. We are looking for the smallest positive solution for $x$ .
3. The solution should be accurate to at least two decimal places.

STEP 2

First, we need to isolate $\sin(2x)$ by dividing both sides of the equation by 6.
$\sin(2x) = \frac{3}{6}$

STEP 3

Now, simplify the fraction on the right-hand side.
$\sin(2x) = \frac{1}{2}$

STEP 4

To find $x$ , we need to take the inverse sine (arcsin) of both sides of the equation. However, since we have $2x$ instead of $x$ , we will first find the angle that corresponds to $\sin(\theta) = \frac{1}{2}$ and then divide that angle by 2 to solve for $x$ .
$2x = \arcsin\left(\frac{1}{2}\right)$

STEP 5

The value of $\arcsin\left(\frac{1}{2}\right)$ is known to be $30^\circ$ or $\frac{\pi}{6}$ radians. We will use radians because they are more common in mathematics for trigonometric functions.
$2x = \frac{\pi}{6}$

STEP 6

Now, divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{\pi}{6} \div 2$

STEP 7

Simplify the right-hand side to find the value of $x$ .
$x = \frac{\pi}{12}$

STEP 8

Convert the radian measure to a decimal to get the smallest positive solution for $x$ .
$x \approx \frac{3.14159}{12}$

STEP 9

Perform the division to obtain the decimal value of $x$ .
$x \approx 0.2618$
The smallest positive solution for $x$ accurate to at least two decimal places is approximately $0.26$ radians.

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Question Find the smallest positive solution to the equation 6sin⁡(2x)=36 \sin (2 x) = 36sin(2x)=3, accurate to at least two decimal places.

Studdy Solution

STEP 1

Assumptions1. We are solving the equation 6sin⁡(2x)=36 \sin (2x) = 36sin(2x)=3.2. We are looking for the smallest positive solution for xxx.3. The solution should be accurate to at least two decimal places.

STEP 2

First, we need to isolate sin⁡(2x)\sin(2x)sin(2x) by dividing both sides of the equation by 6.sin⁡(2x)=36\sin(2x) = \frac{3}{6}sin(2x)=63​

STEP 3

Now, simplify the fraction on the right-hand side.sin⁡(2x)=12\sin(2x) = \frac{1}{2}sin(2x)=21​

STEP 4

STEP 5

The value of arcsin⁡(12)\arcsin\left(\frac{1}{2}\right)arcsin(21​) is known to be 30∘30^\circ30∘ or π6\frac{\pi}{6}6π​ radians. We will use radians because they are more common in mathematics for trigonometric functions.2x=π62x = \frac{\pi}{6}2x=6π​

STEP 6

Now, divide both sides of the equation by 2 to solve for xxx.x=π6÷2x = \frac{\pi}{6} \div 2x=6π​÷2

STEP 7

Simplify the right-hand side to find the value of xxx.x=π12x = \frac{\pi}{12}x=12π​

STEP 8

Convert the radian measure to a decimal to get the smallest positive solution for xxx.x≈3.1415912x \approx \frac{3.14159}{12}x≈123.14159​

STEP 9

Perform the division to obtain the decimal value of xxx.x≈0.2618x \approx 0.2618x≈0.2618The smallest positive solution for xxx accurate to at least two decimal places is approximately 0.260.260.26 radians.

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Question Find the smallest positive solution to the equation $6 \sin (2 x) = 3$ , accurate to at least two decimal places.

Assumptions
1. We are solving the equation $6 \sin (2x) = 3$ .
2. We are looking for the smallest positive solution for $x$ .
3. The solution should be accurate to at least two decimal places.

First, we need to isolate $\sin(2x)$ by dividing both sides of the equation by 6.
$\sin(2x) = \frac{3}{6}$

Now, simplify the fraction on the right-hand side.
$\sin(2x) = \frac{1}{2}$

The value of $\arcsin\left(\frac{1}{2}\right)$ is known to be $30^\circ$ or $\frac{\pi}{6}$ radians. We will use radians because they are more common in mathematics for trigonometric functions.
$2x = \frac{\pi}{6}$

Now, divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{\pi}{6} \div 2$

Simplify the right-hand side to find the value of $x$ .
$x = \frac{\pi}{12}$

Convert the radian measure to a decimal to get the smallest positive solution for $x$ .
$x \approx \frac{3.14159}{12}$

Perform the division to obtain the decimal value of $x$ .
$x \approx 0.2618$
The smallest positive solution for $x$ accurate to at least two decimal places is approximately $0.26$ radians.