0. Functions

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

cos⁻¹ √3/2

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

cos⁻¹ (- 1/2 )

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

sin⁻¹ ( -1 )

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

cos (cos⁻¹ ( -1 ))

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x) = (x−1) sin^−1 x on [−1,1]

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x)=e^−x tan^−1 x on [0,∞)

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>

a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

An inverse tangent identity

b. Prove that tan⁻¹ x + tan⁻¹ x(1/x) = π/2, for x > 0.

Verify the identity sec⁻¹ x = cos⁻¹ (1/x), for x ≠ 0.

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 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.

1. a. arctan 1

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

1. c. tan^(-1)(1/√3)

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

2. b. tan^(-1)(√3)

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

3. a. arcsin(-1/2)