91Ó°ÊÓ

Without the use of a calculator, state the exact value of the trig functions for the given angle. A diagram may help. a. \(\sin \pi\) b. \(\sin 0\) c. \(\sin \left(\frac{\pi}{2}\right)\) d. \(\sin \left(\frac{3 \pi}{2}\right)\)

From Pythagorean triples to points on the unit circle: \((x, y, r) \rightarrow\left(\frac{x}{r}, \frac{y}{r}, 1\right)\) While not strictly a "formula," dividing a Pythagorean triple by \(r\) is a simple algorithm for rewriting any Pythagorean triple as a triple with hypotenuse 1 . This enables us to identify certain points on a unit circle, and to evaluate the six trig functions of the related acute angle. Rewrite each triple as a triple with hypotenuse 1 , verify \(\left(\frac{x}{r}, \frac{y}{r}\right)\) is a point on the unit circle, and evaluate the six trig functions using this point. a. \((5,12,13)\) b. \((7,24,25)\) c. \((12,35,37)\) d. \((9,40,41)\)

Without using a calculator, find the value of \(t\) in \([0,2 \pi)\) that corresponds to the following functions. $$ \sec t=-2 ; t \text { in QIII } $$

Unit circle points: In Exercises 23 through 26, four decimal approximations of unit circle points were given. Find such unit circle points that are on the terminal side of the following angles in standard position. (Hint: Use the definitions of the circular functions.) $$\theta=140^{\circ}$$

Without using a calculator, find the two values of \(t\) (where possible) in \([0,2 \pi\) ) that make each equation true. $$ \csc t=-\frac{2}{\sqrt{3}} $$

Unit circle points: In Exercises 23 through 26, four decimal approximations of unit circle points were given. Find such unit circle points that are on the terminal side of the following angles in standard position. (Hint: Use the definitions of the circular functions.) $$\theta=5$$

Unit circle points: In Exercises 23 through 26, four decimal approximations of unit circle points were given. Find such unit circle points that are on the terminal side of the following angles in standard position. (Hint: Use the definitions of the circular functions.) $$\theta=4$$

Arc length: Both Libreville, Gabon, and Jamame, Somalia, lie near the equator, but on opposite ends of the African continent. If Libreville is at \(9.3^{\circ}\) east longitude and Jamame is \(42.5^{\circ}\) east longitude, how wide is the continent of Africa at the equator?

Using a calculator, find the value of \(t\) in \([0,2 \pi)\) that corresponds to the following functions. Round to four decimal places. $$ \cos t=0.7402, \sin t>0 $$

The planet Neptune has an orbit that is nearly circular. It orbits the Sun at a distance of 4497 million \(\mathrm{km}\) and completes one revolution every \(165 \mathrm{yr}\). (a) Find the angle \(\theta\) that the planet moves through in \(1 \mathrm{yr}\) in both degrees and radians and (b) find the linear velocity \((\mathrm{km} / \mathrm{hr})\) as it orbits the Sun.