0. Functions

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

c. csch⁻¹ 5

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\(\csc\)^{-1}\(\left\)(-1\(\right\))$

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

5. c. arccos(√3/2)

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

f. tan⁻¹(sinh x) |₋₃³

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

4. b. arcsin(-1/√2)

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\(\tan\]\left\)(\(\tan\)^{-1}1\(\right\))$

Evaluate the expression.

$\(\sin\)^{-1}1$

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

2. b. tan^(-1)(√3)

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.

141. a. y=arccos(cos x)

Evaluate the expression.

$\(\tan\)^{-1}\(\left\)(-\(\frac{\sqrt3}{3}\]\right\))$

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.

cos⁻¹ (- 1/2 )

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.

139. a. y=arctan(tan x)

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\(\tan\)^{-1}\(\left\)(\(\tan\[\left\)(\(\frac{3\pi}{4}\]\right\))\(\right\))$

Evaluate the expression using a calculator. Express your answer in radians, rounding to two decimal places.

$\(\cos\)^{-1}\(\left\)(\(\frac\)14\(\right\))$

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

65. sech⁻¹(3/5)