Question

Find the exact value of each expression, if it is defined. Express your answer in radians. (a) sin⁻¹(-2/2) radians (b) cos⁻¹(-1/2) radians (c) tan⁻¹(3/3)

Answer 1

The value of sin⁻¹(-2/2) in radians is approximately -0.524 radians. The value of cos⁻¹(-1/2) in radians is approximately 2.89 radians. The value of tan⁻¹(3/3) in radians is approximately 0.785 radians.

Step-by-step explanation:

(a) To find the value of sin⁻¹(-2/2) in radians, we need to determine the angle whose sine is -2/2. The sine function returns the ratio of the opposite side to the hypotenuse in a right triangle.

For this problem, since the sine is negative, the angle must be in the third or fourth quadrant. The reference angle for sin⁻¹(-2/2) is 30.1°. We convert this angle to radians by multiplying by π/180.

So, sin⁻¹(-2/2) in radians is approximately -0.524 radians.

(b) To find the value of cos⁻¹(-1/2) in radians, we need to determine the angle whose cosine is -1/2. The cosine function returns the ratio of the adjacent side to the hypotenuse in a right triangle.

Since the cosine is negative, the angle must be in the second or third quadrant. The reference angle for cos⁻¹(-1/2) is 48.7°. We convert this angle to radians by multiplying by π/180.

So, cos⁻¹(-1/2) in radians is approximately 2.89 radians.

(c) To find the value of tan⁻¹(3/3) in radians, we need to determine the angle whose tangent is 3/3. The tangent function returns the ratio of the opposite side to the adjacent side in a right triangle.

Since the tangent is positive, the angle must be in the first or third quadrant. The reference angle for tan⁻¹(3/3) is 45°. We convert this angle to radians by multiplying by π/180.

So, tan⁻¹(3/3) in radians is approximately 0.785 radians.

Answer 2

The exact value of each expression is:

`(a) \((-\pi)/(4)\) radians`

`(b) \((2\pi)/(3)\) radians`

`(c) \((\pi)/(4)\) radians`

Step-by-step explanation:

The expression
`\(\sin^(-1)\left((-2)/(2)\right)\)` corresponds to finding the angle whose sine is
`\((-2)/(2)\). The sine function has a range of \([-1, 1]\), and \(\sin^(-1)\)`returns an angle in the interval
`\(\left[-(\pi)/(2), (\pi)/(2)\right]\).` Therefore,
`\(\sin^(-1)\left((-2)/(2)\right) = -(\pi)/(4)\).`

For the expression
`\(\cos^(-1)\left((-1)/(2)\right)\),` it represents finding the angle whose cosine is
`\((-1)/(2)\).` The cosine function has a range of
`\([-1, 1]\), and \(\cos^(-1)\) returns an angle in the interval \([0, \pi]\).` Thus,
`\(\cos^(-1)\left((-1)/(2)\right) = (2\pi)/(3)\).`

Lastly, the expression
`\(\tan^(-1)\left((3)/(3)\right)\)` involves finding the angle whose tangent is
`\((3)/(3)\). The tangent function has a range of \((-\infty, \infty)\), and \(\tan^(-1)\)` returns an angle in the interval
`\(\left(-(\pi)/(2), (\pi)/(2)\right)\). Therefore, \(\tan^(-1)\left((3)/(3)\right) = (\pi)/(4)\).`

In summary, these values represent angles in radians that satisfy the respective trigonometric relationships, and they are within the specified ranges based on the properties of the inverse trigonometric functions.