534 CHAPTER 5 Trigonometric Functions 128. Surveying One student in a surveying class measures an angle as 74.25°,while another student measures the same angle as 74° 20′. Find the difference between these measurements, both to the nearest minute and to the nearest hundredth of a degree. 129. Viewing Field of a Telescope As a consequence of Earth’s rotation, celestial objects such as the moon and the stars appear to move across the sky, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 min in a direction perpendicular to Earth’s axis. 130. Angle Measure of a Star on the American Flag Determine the measure of the angle in each point of the five-pointed star appearing on the American flag. (Hint: Inscribe the star in a circle, and use the following theorem from geometry: An angle whose vertex lies on the circumference of a circle is equal to half the central angle that cuts off the same arc. See the figure.) u 2u u 74.25° Trigonometric Functions To define the six trigonometric functions, we start with an angle u in standard position and choose any point P having coordinates 1x, y2 on the terminal side of angle u. (The point P must not be the vertex of the angle.) See Figure 13. A perpendicular from P to the x-axis at point Q determines a right triangle, having vertices at O, P, and Q. We find the distance r from P1x, y2 to the origin, 10, 02, using the distance formula. d1O, P2 = 21x2 - x122 + 1y 2 - y122 Distance formula r = 21x - 022 + 1y - 022 Substitute 1x, y2 for 1x2, y22 and 10, 02 for 1x1, y12. r =!x2 +y2 Subtract. Notice that r +0 because this is the undirected distance. The six trigonometric functions of angle u are sine, cosine, tangent, cotangent, secant, and cosecant, abbreviated sin, cos, tan, cot, sec, and csc. 5.2 Trigonometric Functions ■ Trigonometric Functions ■ Quadrantal Angles ■ Reciprocal Identities ■ Signs and Ranges of Function Values ■ Pythagorean Identities ■ Quotient Identities x y O Q P(x, y) x y r u Figure 13
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