0. Functions

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

sec (5π/2)

Solve the following equations.

{Use of Tech} $\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\]\lt{\frac{\pi}{2}\)}$

Observers at positions A and B 2 km apart simultaneously measure the angle of elevation of a weather balloon to be 40° and 70°, respectively. If the balloon is directly above a point on the line segment between A and B, find the height of the balloon.

Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.

4. cosh x = 13/5, x>0

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.

f. sinh (2 ln 3)

Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.

9. (sinh(x)+cosh(x))⁴

Which of the following is the period of the function $z=\sin (x-y)$ with respect to $x$?

Solve the following equations.

$\(\sin\)3x=\(\frac{\sqrt{2}\)}{2},0\(\leq{x}\]\lt{2\pi}\)$

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Find the spring’s displacement when t = 0, t = π/3, and t = 3π/4.

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cos (2π/3)

Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal and wire CT makes an angle of 50° with the horizontal, how high is the pole?

57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. If x = 4 tanθ, then cscθ = 4/x.

Solve the following equations.

$\(\tan\)^22\(\theta\)=1,0\(\lt{\theta}\]\lt{\pi}\)$

Given the right triangle below, evaluate $\(\tan\]\theta\)$.

Refer to the given figure. Write the radius r of the circle in terms of α and θ.