Chapter 7: Analytic Trigonometry

Projectile Motion Refer to Problem 109.

(a) If you can throw a baseball with an initial speed of 40 meters per second, at what angle of elevation $\theta$ should you direct the throw so that the ball travels a distance of 110 meters before striking the ground?

(b) Determine the maximum distance that you can throw the ball.

(c) Graph $R=R\left(\theta \right),$with $v_0=40$meters per second.

(d) Verify the results obtained in parts (a) and (b) using a graphing utility.

Show that$\cos \frac{\pi }{8}=\frac{\sqrt{2+\sqrt{2}}}{2}$and use it to find$\sin \frac{\pi }{16},\cos \frac{\pi }{16}$.

Make up an identity that is not a Fundamental Identity.

Discuss the following derivation:

$\tan \left(\theta +\frac{\mathrm{\pi }}{2}\right)=\frac{\tan \left(\theta \right)+\tan \left(\frac{\mathrm{\pi }}{2}\right)}{1-\tan \left(\theta \right)\tan \left(\frac{\mathrm{\pi }}{2}\right)}=\frac{{\displaystyle \frac{\tan \left(\theta \right)}{\tan \left(\frac{\mathrm{\pi }}{2}\right)}}+1}{{\displaystyle \frac{1}{\tan \left(\frac{\mathrm{\pi }}{2}\right)}}-\tan \left(\theta \right)}=\frac{0+1}{0-\tan \left(\theta \right)}=\frac{1}{-\tan \left(\theta \right)}=-cot\left(\theta \right)$

Explain why formula cannot be used to show that

$\tan \left(\frac{\mathrm{\pi }}{2}-\theta \right)=cot\theta$

Establish this identity by using formulas $\left(3a\right)$and$\left(3b\right)$.

Show that${\sin }^3\theta +{\sin }^3\left(\theta +120°\right)+{\sin }^3\left(\theta +240°\right)=-\frac{3}{4}\sin \left(3\theta \right)$

The index of refraction of light in passing from a vacuum into water is 1.33. If the angle of incidence is 40°, determine the angle of refraction.

The index of refraction of light in passing from a vacuum into dense flint glass is 1.66. If the angle of incidence is 50°, determine the angle of refraction.

If $\tan \theta =a\tan \frac{\theta }{3}$, express$\tan \frac{\theta }{3}$in terms of a.

Ptolemy, who lived in the city of Alexandria in Egypt during the second century ad, gave the measured values in the following table for the angle of incidence ${\theta }_1$and the angle of refraction ${\theta }_2$for a light beam passing from air into water. Do these values agree with Snell’s Law? If so, what index of refraction results? (These data are of interest as the oldest recorded physical measurements.)