Chapter 7: Analytic Trigonometry

Bending Light The speed of yellow sodium light (wavelength, $589$nano-meters) in a certain liquid is measured to be $1.92\times {10}^8$meters per second. What is the index of refraction of this liquid, with respect to air, for sodium light?

[Hint: The speed of light in air is approximately $2.998\times {10}^8$meters per second.]

Bending Light A beam of light with a wavelength of 589 nano-meters traveling in air makes an angle of incidence of 40° on a slab of transparent material, and the refracted beam makes an angle of refraction of 26°. Find the index of refraction of the material.

Bending Light A light ray with a wavelength of 589 nano-meters (produced by a sodium lamp) traveling through air makes an angle of incidence of 30° on a smooth, flat slab of crown glass. Find the angle of refraction.

A light beam passes through a thick slab of material whose index of refraction is $n_2$. Show that the emerging beam is parallel to the incident beam.

Brewster’s Law If the angle of incidence and the angle of refraction are complementary angles, the angle of incidence is referred to as the Brewster angle ${\theta }_B.$The Brewster angle is related to the index of refractions of the two media, $n_1\mathrm{and}{n}_2,$by the equation $n_1\sin {\theta }_B=n_2\cos {\theta }_B,$where $n_1$ is the index of refraction of the incident medium and$n_2$ is the index of refraction of the refractive medium. Determine the Brewster angle for a light beam traveling through water (at 20°C) that makes an angle of incidence with a smooth, flat slab of crown glass.

Explain in your own words how you would use your calculator to solve the equation $\cos x=-0.6,0\le x\le 2\mathrm{\pi }$. How would you modify your approach to solve the equation $\cot x=-0.6,0\le x\le 2\mathrm{\pi }$?

In Problems 9–36, find the exact value of each expression.

$\tan \left[{\sin }^{-1}\left(-\frac{1}{2}\right)\right]$

Multiply:$\frac{\sin \left(\theta \right)}{1+\cos \left(\theta \right)}\frac{1-\cos \left(\theta \right)}{1-\cos \left(\theta \right)}$

Express the product $\sin 4\theta \cos 6\theta$as a sum containing only sines or only cosines.

True or False$y={\tan }^{-1}xmeansx=\tan y,where-\infty <x<\infty and-\frac{\mathrm{\pi }}{2}<y<\frac{\mathrm{\pi }}{2}$