QuestionFind the exact value of $\csc^{-1}(-2)$ . Options: $\frac{2 \pi}{3}$ , $-\frac{\pi}{6}$ , $\frac{\pi}{6}$ , $\frac{\pi}{3}$ .

Studdy Solution

STEP 1

Assumptions1. The problem asks for the exact value of the inverse cosecant of -, denoted as $\csc^{-1}(-)$ . . The range of $\csc^{-1}(x)$ is $(-\pi/,0] \cup [0, \pi/)$ .
3. The domain of $\csc^{-1}(x)$ is $(-\infty, -1] \cup [1, \infty)$ .

STEP 2

We know that $\csc^{-1}(x)$ is the angle whose cosecant is $x$ . So, we are looking for an angle whose cosecant is -2.

STEP 3

The cosecant of an angle is defined as the reciprocal of the sine of the angle. So, we can write the given expression as $\sin^{-1}(-1/2)$ .
$\csc^{-1}(-2) = \sin^{-1}(-1/2)$

STEP 4

Now, we need to find an angle in the range of the inverse sine function whose sine is -1/2.

STEP 5

We know that $\sin(-\pi/) = -1/2$ . Therefore, $\sin^{-1}(-1/2) = -\pi/$

STEP 6

So, the exact value of the expression $\csc^{-1}(-2)$ is $-\pi/6$ .
The correct answer is $-\frac{\pi}{6}$ .

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QuestionFind the exact value of $\csc^{-1}(-2)$ . Options: $\frac{2 \pi}{3}$ , $-\frac{\pi}{6}$ , $\frac{\pi}{6}$ , $\frac{\pi}{3}$ .

Studdy Solution

STEP 1

Assumptions1. The problem asks for the exact value of the inverse cosecant of -, denoted as $\csc^{-1}(-)$ . . The range of $\csc^{-1}(x)$ is $(-\pi/,0] \cup [0, \pi/)$ .
3. The domain of $\csc^{-1}(x)$ is $(-\infty, -1] \cup [1, \infty)$ .

STEP 2

We know that $\csc^{-1}(x)$ is the angle whose cosecant is $x$ . So, we are looking for an angle whose cosecant is -2.

STEP 3

The cosecant of an angle is defined as the reciprocal of the sine of the angle. So, we can write the given expression as $\sin^{-1}(-1/2)$ .
$\csc^{-1}(-2) = \sin^{-1}(-1/2)$

STEP 4

Now, we need to find an angle in the range of the inverse sine function whose sine is -1/2.

STEP 5

We know that $\sin(-\pi/) = -1/2$ . Therefore, $\sin^{-1}(-1/2) = -\pi/$

STEP 6

So, the exact value of the expression $\csc^{-1}(-2)$ is $-\pi/6$ .
The correct answer is $-\frac{\pi}{6}$ .

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QuestionFind the exact value of csc⁡−1(−2)\csc^{-1}(-2)csc−1(−2). Options: 2π3\frac{2 \pi}{3}32π​, −π6-\frac{\pi}{6}−6π​, π6\frac{\pi}{6}6π​, π3\frac{\pi}{3}3π​.

Studdy Solution

STEP 1

STEP 2

We know that csc⁡−1(x)\csc^{-1}(x)csc−1(x) is the angle whose cosecant is xxx. So, we are looking for an angle whose cosecant is -2.

STEP 3

The cosecant of an angle is defined as the reciprocal of the sine of the angle. So, we can write the given expression as sin⁡−1(−1/2)\sin^{-1}(-1/2)sin−1(−1/2).csc⁡−1(−2)=sin⁡−1(−1/2)\csc^{-1}(-2) = \sin^{-1}(-1/2)csc−1(−2)=sin−1(−1/2)

STEP 4

Now, we need to find an angle in the range of the inverse sine function whose sine is -1/2.

STEP 5

We know that sin⁡(−π/)=−1/2\sin(-\pi/) = -1/2sin(−π/)=−1/2. Therefore,sin⁡−1(−1/2)=−π/\sin^{-1}(-1/2) = -\pi/sin−1(−1/2)=−π/

STEP 6

So, the exact value of the expression csc⁡−1(−2)\csc^{-1}(-2)csc−1(−2) is −π/6-\pi/6−π/6.The correct answer is −π6-\frac{\pi}{6}−6π​.

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QuestionFind the exact value of $\csc^{-1}(-2)$ . Options: $\frac{2 \pi}{3}$ , $-\frac{\pi}{6}$ , $\frac{\pi}{6}$ , $\frac{\pi}{3}$ .

We know that $\csc^{-1}(x)$ is the angle whose cosecant is $x$ . So, we are looking for an angle whose cosecant is -2.

The cosecant of an angle is defined as the reciprocal of the sine of the angle. So, we can write the given expression as $\sin^{-1}(-1/2)$ .
$\csc^{-1}(-2) = \sin^{-1}(-1/2)$

We know that $\sin(-\pi/) = -1/2$ . Therefore, $\sin^{-1}(-1/2) = -\pi/$

So, the exact value of the expression $\csc^{-1}(-2)$ is $-\pi/6$ .
The correct answer is $-\frac{\pi}{6}$ .