91Ó°ÊÓ

Find \(\theta\) to four significant digits for \(0 \leq \theta<2 \pi\). $$\sec \theta=-1.307$$

$$\text {Evaluate the given expressions.}$$ The force \(F\) that a rope exerts on a crate is related to force \(F_{x}\) directed along the \(x\) -axis by \(F=F_{x} \sec \theta\) where \(\theta\) is the standard-position angle for \(F\). See Fig. \(8.18 .\) Find \(F\) if \(F_{x}=-365 \mathrm{N}\) and \(\theta=127.0^{\circ}\)

Find \(\theta\) to four significant digits for \(0 \leq \theta<2 \pi\). $$\csc \theta=3.940$$

Another use of radians is illustrated. Use a calculator (in radian mode) to evaluate the ratios \((\sin \theta) / \theta\) and \((\tan \theta) / \theta\) for \(\theta=0.1,0.01,0.001,\) and \(0.0001 .\) From these values, explain why it is possible to say that $$\sin \theta=\tan \theta=\theta$$ approximately for very small angles measured in radians.

Solve the given problems. (Hint: For problems 61-64, review cofunctions on page 125.) Find the radian measure of an angle at the center of a circle of radius \(12 \mathrm{cm}\) that intercepts an arc of \(15 \mathrm{cm}\) on the circle.

Solve the given problems. (Hint: For problems 61-64, review cofunctions on page 125.) Find the length of arc of a circle of radius 10 in. that is intercepted from the center of the circle by an angle of 3 radians.

Solve the given problems. (Hint: For problems 61-64, review cofunctions on page 125.) Using the fact that \(\tan \frac{\pi}{6}=0.5774,\) find the value of \(\cot \frac{5 \pi}{3}\). (A calculator should be used only to check the result.)

Solve the given problems. (Hint: For problems 61-64, review cofunctions on page 125.) Express \(\tan \left(\frac{\pi}{2}+\theta\right)\) in terms of \(\cot \theta .\left(0<\theta<\frac{\pi}{2}\right)\)

Solve the given problems. (Hint: For problems 61-64, review cofunctions on page 125.) Express \(\cos \left(\frac{3 \pi}{2}+\theta\right)\) in terms of \(\sin \theta .\left(0<\theta<\frac{\pi}{2}\right)\)

Evaluate the given problems. A unit of angle measurement used in artillery is the mil, which is defined as a central angle of a circle that intercepts an arc equal in length to \(1 / 6400\) of the circumference. How many mils are in a central angle of \(34.4^{\circ} ?\)