Solved on Feb 07, 2024

Evaluate the expression $\sin x \csc (-x)$ .

STEP 1

Assumptions
1. We are working with trigonometric functions.
2. The variable $x$ represents an angle, which can be in degrees or radians.
3. The trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$ will be used, where $\theta$ is an angle.

STEP 2

We need to simplify the expression $\sin x \csc (-x)$ . To do this, we will use the trigonometric identity for cosecant.
$\csc(\theta) = \frac{1}{\sin(\theta)}$

STEP 3

We will also use the fact that sine is an odd function, which means that $\sin(-\theta) = -\sin(\theta)$ .

STEP 4

Now, we substitute the identity for $\csc(-x)$ in terms of $\sin(-x)$ .
$\sin x \csc (-x) = \sin x \cdot \frac{1}{\sin(-x)}$

STEP 5

Next, we apply the property of the sine function being odd to $\sin(-x)$ .
$\sin(-x) = -\sin(x)$

STEP 6

Substitute the result from STEP_5 into the expression.
$\sin x \cdot \frac{1}{\sin(-x)} = \sin x \cdot \frac{1}{-\sin(x)}$

STEP 7

We can now simplify the expression by canceling out $\sin x$ in the numerator with $\sin x$ in the denominator, keeping in mind that we have a negative sign in the denominator.
$\sin x \cdot \frac{1}{-\sin(x)} = \frac{\sin x}{-\sin x}$

STEP 8

The $\sin x$ terms cancel out, leaving us with the final simplified result.
$\frac{\sin x}{-\sin x} = -1$
The simplified result of $\sin x \csc (-x)$ is $-1$ .

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Evaluate the expression $\sin x \csc (-x)$ .

STEP 1

Assumptions
1. We are working with trigonometric functions.
2. The variable $x$ represents an angle, which can be in degrees or radians.
3. The trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$ will be used, where $\theta$ is an angle.

STEP 2

We need to simplify the expression $\sin x \csc (-x)$ . To do this, we will use the trigonometric identity for cosecant.
$\csc(\theta) = \frac{1}{\sin(\theta)}$

STEP 3

We will also use the fact that sine is an odd function, which means that $\sin(-\theta) = -\sin(\theta)$ .

STEP 4

Now, we substitute the identity for $\csc(-x)$ in terms of $\sin(-x)$ .
$\sin x \csc (-x) = \sin x \cdot \frac{1}{\sin(-x)}$

STEP 5

Next, we apply the property of the sine function being odd to $\sin(-x)$ .
$\sin(-x) = -\sin(x)$

STEP 6

Substitute the result from STEP_5 into the expression.
$\sin x \cdot \frac{1}{\sin(-x)} = \sin x \cdot \frac{1}{-\sin(x)}$

STEP 7

We can now simplify the expression by canceling out $\sin x$ in the numerator with $\sin x$ in the denominator, keeping in mind that we have a negative sign in the denominator.
$\sin x \cdot \frac{1}{-\sin(x)} = \frac{\sin x}{-\sin x}$

STEP 8

The $\sin x$ terms cancel out, leaving us with the final simplified result.
$\frac{\sin x}{-\sin x} = -1$
The simplified result of $\sin x \csc (-x)$ is $-1$ .

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Evaluate the expression sin⁡xcsc⁡(−x)\sin x \csc (-x)sinxcsc(−x).

STEP 1

Assumptions1. We are working with trigonometric functions.2. The variable xxx represents an angle, which can be in degrees or radians.3. The trigonometric identity csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}csc(θ)=sin(θ)1​ will be used, where θ\thetaθ is an angle.

STEP 2

We need to simplify the expression sin⁡xcsc⁡(−x)\sin x \csc (-x)sinxcsc(−x). To do this, we will use the trigonometric identity for cosecant.csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}csc(θ)=sin(θ)1​

STEP 3

We will also use the fact that sine is an odd function, which means that sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)sin(−θ)=−sin(θ).

STEP 4

Now, we substitute the identity for csc⁡(−x)\csc(-x)csc(−x) in terms of sin⁡(−x)\sin(-x)sin(−x).sin⁡xcsc⁡(−x)=sin⁡x⋅1sin⁡(−x)\sin x \csc (-x) = \sin x \cdot \frac{1}{\sin(-x)}sinxcsc(−x)=sinx⋅sin(−x)1​

STEP 5

Next, we apply the property of the sine function being odd to sin⁡(−x)\sin(-x)sin(−x).sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x)sin(−x)=−sin(x)

STEP 6

Substitute the result from STEP_5 into the expression.sin⁡x⋅1sin⁡(−x)=sin⁡x⋅1−sin⁡(x)\sin x \cdot \frac{1}{\sin(-x)} = \sin x \cdot \frac{1}{-\sin(x)}sinx⋅sin(−x)1​=sinx⋅−sin(x)1​

STEP 7

STEP 8

The sin⁡x\sin xsinx terms cancel out, leaving us with the final simplified result.sin⁡x−sin⁡x=−1\frac{\sin x}{-\sin x} = -1−sinxsinx​=−1The simplified result of sin⁡xcsc⁡(−x)\sin x \csc (-x)sinxcsc(−x) is −1-1−1.

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Evaluate the expression $\sin x \csc (-x)$ .

Assumptions
1. We are working with trigonometric functions.
2. The variable $x$ represents an angle, which can be in degrees or radians.
3. The trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$ will be used, where $\theta$ is an angle.

We need to simplify the expression $\sin x \csc (-x)$ . To do this, we will use the trigonometric identity for cosecant.
$\csc(\theta) = \frac{1}{\sin(\theta)}$

We will also use the fact that sine is an odd function, which means that $\sin(-\theta) = -\sin(\theta)$ .

Now, we substitute the identity for $\csc(-x)$ in terms of $\sin(-x)$ .
$\sin x \csc (-x) = \sin x \cdot \frac{1}{\sin(-x)}$

Next, we apply the property of the sine function being odd to $\sin(-x)$ .
$\sin(-x) = -\sin(x)$

Substitute the result from STEP_5 into the expression.
$\sin x \cdot \frac{1}{\sin(-x)} = \sin x \cdot \frac{1}{-\sin(x)}$

The $\sin x$ terms cancel out, leaving us with the final simplified result.
$\frac{\sin x}{-\sin x} = -1$
The simplified result of $\sin x \csc (-x)$ is $-1$ .