Concept explainers
Shadow length
The length
(a) Verify that the expression for
(b) Use a graphing utility to create a table of the lengths
(c) Use your table from part (b) to determine the angle of the sun that results in the minimum length of the shadow.
(d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is
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Chapter 2 Solutions
Trigonometry (MindTap Course List)
- Vibrating String When a violin string vibrates, the sound produced results from a combination of standing waves that have evenly placed nodes. The figure (1) illustrates some of the possible standing waves. Lets assume that the string has length . a.For fixed t, the string has the shape of a sine curve y=Asinx. Find the appropriate value of for each of the illustrated standing waves. b.Do you notice a pattern in the values of that you found in part a? What would the next two values of be? Sketch rough graphs of the standing waves associated with these new values of . c.Suppose that for fixed t, each point on the string that is not a node vibrates with frequency 440Hz. Find the value of for which an equation of the form y=Acost would model this motion. d.Combine your answers for parts a and c to find functions of the form y(x,t)=Asinxcost that model each of the standing waves in the figure. Assume that A=1. Figure (1)arrow_forwardTotal Internal Reflection When light passes from a more dense to a less dense medium-from glass to air, for example-the angle of refraction predicted by Snells Law see exercise 57 can be 90 or large. In this case the light beam is actually reflected Bach into the denser medium. This phenomenon, called total internal reflection, is the principal behind optics. Set 2=90 in Snells Law, and solve for 1 to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. Note that the index of refraction from glass to air is reciprocal of the index from air to glass.arrow_forwardHyperbolic sine Function The hyperbolic sine function is defined by sinh(x)=exex2 a Sketch the graph of this function using graphical addition as in Exercise 17. b Use the definition to show that sinh(x)=sinh(x) Hyperbolic Cosine Function The hyperbolic cosine function is defined by cosh(x)=ex+ex2 a Sketch the graphs of the functions y=13ex and y=12ex on the same axes, and use graphical addition see Section 2.7 to sketch the graph of y=cosh(x). b Use the definition to show that cosh(x)=cosh(x).arrow_forward
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