Math / Trigonometry

QuestionSolve $\sin (2 x) \cos (5 x)-\cos (2 x) \sin (5 x)=-0.4$ for the smallest positive solution. $x=$
Give your answer accurate to two decimal places.

Studdy Solution

STEP 1

1. The equation involves trigonometric identities.
2. We will use the sine difference identity to simplify the equation.
3. We are looking for the smallest positive solution for $x$ .

STEP 2

1. Apply the sine difference identity.
2. Simplify the equation.
3. Solve for $x$ .
4. Find the smallest positive solution.

STEP 3

Apply the sine difference identity. The identity states:
$\sin(a)\cos(b) - \cos(a)\sin(b) = \sin(a-b)$
In this case, let $a = 2x$ and $b = 5x$ . Thus, the equation becomes:
$\sin(2x)\cos(5x) - \cos(2x)\sin(5x) = \sin((2x) - (5x)) = \sin(-3x)$

STEP 4

The equation simplifies to:
$\sin(-3x) = -0.4$
Using the property $\sin(-\theta) = -\sin(\theta)$ , we have:
$-\sin(3x) = -0.4$
Thus:
$\sin(3x) = 0.4$

STEP 5

Solve for $3x$ using the inverse sine function:
$3x = \arcsin(0.4)$
Calculate $\arcsin(0.4)$ :
$3x \approx 0.4115 \text{ radians}$

STEP 6

Since the sine function is periodic with period $2\pi$ , the general solution for $3x$ is:
$3x = 0.4115 + 2k\pi \quad \text{or} \quad 3x = \pi - 0.4115 + 2k\pi$
where $k$ is an integer.

STEP 7

Solve for $x$ by dividing by 3:
For the first solution:
$x = \frac{0.4115 + 2k\pi}{3}$
For the second solution:
$x = \frac{\pi - 0.4115 + 2k\pi}{3}$
Find the smallest positive $x$ :
For $k = 0$ :
$x \approx \frac{0.4115}{3} \approx 0.1372$
For the second solution with $k = 0$ :
$x \approx \frac{\pi - 0.4115}{3} \approx 0.9108$
The smallest positive solution is:
$x \approx 0.14$
The smallest positive solution for $x$ is:
$\boxed{0.14}$

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QuestionSolve $\sin (2 x) \cos (5 x)-\cos (2 x) \sin (5 x)=-0.4$ for the smallest positive solution. $x=$
Give your answer accurate to two decimal places.

Studdy Solution

STEP 1

1. The equation involves trigonometric identities.
2. We will use the sine difference identity to simplify the equation.
3. We are looking for the smallest positive solution for $x$ .

STEP 2

1. Apply the sine difference identity.
2. Simplify the equation.
3. Solve for $x$ .
4. Find the smallest positive solution.

STEP 3

Apply the sine difference identity. The identity states:
$\sin(a)\cos(b) - \cos(a)\sin(b) = \sin(a-b)$
In this case, let $a = 2x$ and $b = 5x$ . Thus, the equation becomes:
$\sin(2x)\cos(5x) - \cos(2x)\sin(5x) = \sin((2x) - (5x)) = \sin(-3x)$

STEP 4

The equation simplifies to:
$\sin(-3x) = -0.4$
Using the property $\sin(-\theta) = -\sin(\theta)$ , we have:
$-\sin(3x) = -0.4$
Thus:
$\sin(3x) = 0.4$

STEP 5

Solve for $3x$ using the inverse sine function:
$3x = \arcsin(0.4)$
Calculate $\arcsin(0.4)$ :
$3x \approx 0.4115 \text{ radians}$

STEP 6

Since the sine function is periodic with period $2\pi$ , the general solution for $3x$ is:
$3x = 0.4115 + 2k\pi \quad \text{or} \quad 3x = \pi - 0.4115 + 2k\pi$
where $k$ is an integer.

STEP 7

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QuestionSolve sin⁡(2x)cos⁡(5x)−cos⁡(2x)sin⁡(5x)=−0.4\sin (2 x) \cos (5 x)-\cos (2 x) \sin (5 x)=-0.4sin(2x)cos(5x)−cos(2x)sin(5x)=−0.4 for the smallest positive solution. x=x=x=Give your answer accurate to two decimal places.

Studdy Solution

STEP 1

1. The equation involves trigonometric identities.2. We will use the sine difference identity to simplify the equation.3. We are looking for the smallest positive solution for x x x.

STEP 2

1. Apply the sine difference identity.2. Simplify the equation.3. Solve for x x x.4. Find the smallest positive solution.

STEP 3

STEP 4

The equation simplifies to:sin⁡(−3x)=−0.4\sin(-3x) = -0.4sin(−3x)=−0.4Using the property sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)sin(−θ)=−sin(θ), we have:−sin⁡(3x)=−0.4-\sin(3x) = -0.4−sin(3x)=−0.4Thus:sin⁡(3x)=0.4\sin(3x) = 0.4sin(3x)=0.4

STEP 5

Solve for 3x 3x 3x using the inverse sine function:3x=arcsin⁡(0.4)3x = \arcsin(0.4)3x=arcsin(0.4)Calculate arcsin⁡(0.4)\arcsin(0.4)arcsin(0.4):3x≈0.4115 radians3x \approx 0.4115 \text{ radians}3x≈0.4115 radians

STEP 6

Since the sine function is periodic with period 2π2\pi2π, the general solution for 3x3x3x is:3x=0.4115+2kπor3x=π−0.4115+2kπ3x = 0.4115 + 2k\pi \quad \text{or} \quad 3x = \pi - 0.4115 + 2k\pi3x=0.4115+2kπor3x=π−0.4115+2kπwhere kkk is an integer.

STEP 7

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Studdy solves anything!

Start learning now

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QuestionSolve $\sin (2 x) \cos (5 x)-\cos (2 x) \sin (5 x)=-0.4$ for the smallest positive solution. $x=$
Give your answer accurate to two decimal places.

1. The equation involves trigonometric identities.
2. We will use the sine difference identity to simplify the equation.
3. We are looking for the smallest positive solution for $x$ .

1. Apply the sine difference identity.
2. Simplify the equation.
3. Solve for $x$ .
4. Find the smallest positive solution.

The equation simplifies to:
$\sin(-3x) = -0.4$
Using the property $\sin(-\theta) = -\sin(\theta)$ , we have:
$-\sin(3x) = -0.4$
Thus:
$\sin(3x) = 0.4$

Solve for $3x$ using the inverse sine function:
$3x = \arcsin(0.4)$
Calculate $\arcsin(0.4)$ :
$3x \approx 0.4115 \text{ radians}$

Since the sine function is periodic with period $2\pi$ , the general solution for $3x$ is:
$3x = 0.4115 + 2k\pi \quad \text{or} \quad 3x = \pi - 0.4115 + 2k\pi$
where $k$ is an integer.