Solved on Feb 19, 2024

Find the angle whose cosine is $-\sqrt{2}/2$ .

STEP 1

Assumptions
1. We need to find the value of the inverse cosine function, $\cos^{-1}$ , for the argument $-\frac{\sqrt{2}}{2}$ .
2. The range of the principal value for $\cos^{-1}x$ is $[0, \pi]$ in radians.

STEP 2

Recall the definition of the inverse cosine function, $\cos^{-1}$ . It gives the angle whose cosine is the given number.
$\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \theta \implies \cos(\theta) = -\frac{\sqrt{2}}{2}$

STEP 3

Recognize that the cosine of an angle is negative in the second and third quadrants of the unit circle.

STEP 4

Recall the special angles and their cosine values. The cosine of $\frac{\pi}{4}$ (or 45 degrees) is $\frac{\sqrt{2}}{2}$ .

STEP 5

Since the cosine value we are looking for is negative, we need to find the angle in the second quadrant that corresponds to the positive cosine value in the first quadrant.

STEP 6

The angle in the second quadrant that has the same cosine value (ignoring the sign) as $\frac{\pi}{4}$ is $\pi - \frac{\pi}{4}$ .

STEP 7

Calculate the angle in the second quadrant.
$\theta = \pi - \frac{\pi}{4}$

STEP 8

Simplify the expression to find the value of $\theta$ .
$\theta = \frac{4\pi}{4} - \frac{\pi}{4}$

STEP 9

Combine the terms to get the final value of $\theta$ .
$\theta = \frac{3\pi}{4}$
Therefore, $\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$ .

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Find the angle whose cosine is $-\sqrt{2}/2$ .

STEP 1

Assumptions
1. We need to find the value of the inverse cosine function, $\cos^{-1}$ , for the argument $-\frac{\sqrt{2}}{2}$ .
2. The range of the principal value for $\cos^{-1}x$ is $[0, \pi]$ in radians.

STEP 2

Recall the definition of the inverse cosine function, $\cos^{-1}$ . It gives the angle whose cosine is the given number.
$\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \theta \implies \cos(\theta) = -\frac{\sqrt{2}}{2}$

STEP 3

Recognize that the cosine of an angle is negative in the second and third quadrants of the unit circle.

STEP 4

Recall the special angles and their cosine values. The cosine of $\frac{\pi}{4}$ (or 45 degrees) is $\frac{\sqrt{2}}{2}$ .

STEP 5

Since the cosine value we are looking for is negative, we need to find the angle in the second quadrant that corresponds to the positive cosine value in the first quadrant.

STEP 6

The angle in the second quadrant that has the same cosine value (ignoring the sign) as $\frac{\pi}{4}$ is $\pi - \frac{\pi}{4}$ .

STEP 7

Calculate the angle in the second quadrant.
$\theta = \pi - \frac{\pi}{4}$

STEP 8

Simplify the expression to find the value of $\theta$ .
$\theta = \frac{4\pi}{4} - \frac{\pi}{4}$

STEP 9

Combine the terms to get the final value of $\theta$ .
$\theta = \frac{3\pi}{4}$
Therefore, $\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$ .

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Find the angle whose cosine is −2/2-\sqrt{2}/2−2​/2.

STEP 1

Assumptions1. We need to find the value of the inverse cosine function, cos⁡−1\cos^{-1}cos−1, for the argument −22-\frac{\sqrt{2}}{2}−22​​.2. The range of the principal value for cos⁡−1x\cos^{-1}xcos−1x is [0,π][0, \pi][0,π] in radians.

STEP 2

STEP 3

Recognize that the cosine of an angle is negative in the second and third quadrants of the unit circle.

STEP 4

Recall the special angles and their cosine values. The cosine of π4\frac{\pi}{4}4π​ (or 45 degrees) is 22\frac{\sqrt{2}}{2}22​​.

STEP 5

Since the cosine value we are looking for is negative, we need to find the angle in the second quadrant that corresponds to the positive cosine value in the first quadrant.

STEP 6

The angle in the second quadrant that has the same cosine value (ignoring the sign) as π4\frac{\pi}{4}4π​ is π−π4\pi - \frac{\pi}{4}π−4π​.

STEP 7

Calculate the angle in the second quadrant.θ=π−π4\theta = \pi - \frac{\pi}{4}θ=π−4π​

STEP 8

Simplify the expression to find the value of θ\thetaθ.θ=4π4−π4\theta = \frac{4\pi}{4} - \frac{\pi}{4}θ=44π​−4π​

STEP 9

Combine the terms to get the final value of θ\thetaθ.θ=3π4\theta = \frac{3\pi}{4}θ=43π​Therefore, cos⁡−1(−22)=3π4\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}cos−1(−22​​)=43π​.

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Find the angle whose cosine is $-\sqrt{2}/2$ .

Assumptions
1. We need to find the value of the inverse cosine function, $\cos^{-1}$ , for the argument $-\frac{\sqrt{2}}{2}$ .
2. The range of the principal value for $\cos^{-1}x$ is $[0, \pi]$ in radians.

Recall the special angles and their cosine values. The cosine of $\frac{\pi}{4}$ (or 45 degrees) is $\frac{\sqrt{2}}{2}$ .

The angle in the second quadrant that has the same cosine value (ignoring the sign) as $\frac{\pi}{4}$ is $\pi - \frac{\pi}{4}$ .

Calculate the angle in the second quadrant.
$\theta = \pi - \frac{\pi}{4}$

Simplify the expression to find the value of $\theta$ .
$\theta = \frac{4\pi}{4} - \frac{\pi}{4}$

Combine the terms to get the final value of $\theta$ .
$\theta = \frac{3\pi}{4}$
Therefore, $\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$ .