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The circle in the figure has diameter 1. What are the values of a. b, and c? (Note: This result provides an alternative way to define the sine
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Trigonometry
- Refraction When you stand in shallow water and look at an object below the surface of the water, the object will look farther away from you than it really is. This is because when light rays pass between air and water, the water refracts, or bends, the light rays. The index of refraction for water is 1.333. This is the ratio of the sine of 1 and the sine of 2 (see figure). (a) While standing in water that is 2 feet deep, you look at a rock at angle 1=60 (measured from a line perpendicular to the surface of the water). Find 2. (b) Find the distances x and y. (c) Find the distance d between where the rock is and where it appears to be. (d) What happens to d as you move closer to the rock? Explain.arrow_forwardDistance to the moon To find the distance to the sun as in Exercise 67, we needed to know the distance to the moon. Here is a way to estimate the distance When the moon is seen at its zenith at a point A on the earth, it is observed to be at the horizon from point B see the following figure. Points A and B are 6155 mi apart, and the radius of the earth is 3960 mi. a Find the angle in degree. b Estimate the distance from point A to the moon.arrow_forwardSimilar Triangles and Trigonometric Functions Use the figure below. (a) Explain why ABC,ADE, and AFG are similar triangles. (b) What does similarity imply about the ratios BCAB,DEAD,andFGAF? (c) Does the value of sinA depend on which triangle from part a is used to calculate it? Does the value of sinA change when you use a different right triangle similar to the three given triangles? (d) Do your conclusions from part c apply to the other five trigonometric functions? Explain.arrow_forward
- a Let an acute angle in a right triangle. Identify the opposite side, the adjacent side, and the hypotenuse in the figure. b Define the six trigonometric ratios in terms of the adjacent and opposite sides and the hypotenuse. c Find the six trigonometric ratios for the angle shown in the figure. d List the special values of sine, cosine and tangent.arrow_forwardLength of a Bisector In triangle ABC see the figure the line segment s bisects angle C. Show that the length of s is given by s=2abcosxa+b Hint: Use the Law of Sines.arrow_forward3-8. Finding an Angle or Side Use the Law of Sines to find the indicated side x or angle .arrow_forward
- a Define the inverse sine function, the inverse cosine function, and the inverse tangent function. b Find sin-112,cos-122, and tan-11. c For what values of x is the equation sinsin-1x=x true? For what values of x is the equation sin-1sinx=x true?arrow_forwardFrequency With simple harmonic motion, the reciprocal of the period is called the frequency. The frequency, given by f=1/period represents the number of cycles (or oscillations) that are completed per unit time. The units used to describe frequency are Hertz, where 1HZ=1 cycle per second. Alternating Current In Europe, the voltage of the alternating current coming through an electrical outlet can be modeled by the function V=230sin(100t), where t is measured in seconds and V in volts. What is the frequency of the voltage?arrow_forwardGeometry Use a compass to sketch a quarter of a circle of radius 10 centimeters. Using a protractor, construct an angle of 20 in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. By actual measurement, calculate the coordinates x,y of the point of intersection and use these measurements to approximate the six trigonometric functions of a 20 angle.arrow_forward
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