0. Functions

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

1. c. tan^(-1)(1/√3)

Evaluate the expression.

$\(\cos\]\left\)(\(\cos\)^{-1}\(\left\)(-\(\sqrt\)3\(\right\))\(\right\))$

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

$\(\tan\)^{-1}\(\sqrt\)3$

Evaluate the expression.

$\(\cos\)^{-1}\(\left\)(\(\cos\[\left\)(\(\frac{\pi}{2}\]\right\))\(\right\))$

Find the values in Exercises 9–12.

11. tan(arcsin(-1/2))

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.

63. tanh⁻¹(-1/2)

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

3. a. arcsin(-1/2)

Evaluate the expression.

$\(\sin\)^{-1}\(\left\)(\(\frac{\sqrt3}{2}\]\right\))$

Evaluate the expression.

$\(\tan\)^{-1}0$

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

7. a. sec^(-1)(-√2)

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

7. c. arcsec(-2)

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

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

Evaluate the expression.

$\(\cos\)^{-1}\(\left\)(0\(\right\))$

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

sin⁻¹ ( -1 )

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

3. c. sin^(-1)(-√3/2)