Chapter 7: Analytic Trigonometry

Graph $f\left(x\right)={\sin }^{2}x=\frac{1-\cos \left(2x\right)}{2}$for $0\le x\le 2\pi$by hand using transformations.

Heat Transfer In the study of heat transfer, the equation $x+\tan x=0$occurs. Graph $Y_1=-x$and $Y_2=\tan x$for role="math" localid="1647098211924" $x\ge 0$. Conclude that there are an infinite number of points of intersection of these two graphs. Now find the first two positive solutions of $x+\tan x=0$rounded to two decimal places.

Carrying a Ladder Around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of $\theta$is $L\left(\theta \right)=4csc\theta +3sec\theta .$

(a) In calculus, you are asked to find the length of the longest ladder that can turn the corner by solving the equation

$3sec\theta \tan \theta -4csc\theta cot\theta =0,0°<\theta <90°$

Solve this equation for $\theta .$

(b) What is the length of the longest ladder that can be carried around the corner?

(c) Graph $L=L\left(\theta \right),0°\le \theta \le 90°,$and find the angle $\theta$ that minimizes the length L.

(d) Compare the result with the one found in part (b). Explain why the two answers are the same.

Repeat Problem $107$for$g\left(x\right)={\cos }^{2}x$.

Write down the three Pythagorean Identities.

If $\alpha +\beta +\gamma ={180}^{\circ }$and $cot\theta =cot\alpha +cot\beta +cot\gamma ,0<\theta <{180}^{\circ }$

Show that${\sin }^3\theta =\sin (\alpha -\theta )\sin (\beta -\theta )\sin (\gamma -\theta )$

Use the fact that

$\cos \frac{\pi }{12}=\frac{1}{4}\left(\sqrt{6}+\sqrt{2}\right)$

to find$\sin \frac{\mathrm{\pi }}{24},\cos \frac{\mathrm{\pi }}{24}$.

Why do you think it is usually preferable to start with the side containing the more complicated expression when establishing an identity?

Projectile Motion The horizontal distance that a projectile will

travel in the air (ignoring air resistance) is given by the equation

$R\left(\theta \right)=\frac{{v_0}^2\sin \left(2\theta \right)}{g}$

where $v_0$is the initial velocity of the projectile, $\theta$is the angle

of elevation, and g is acceleration due to gravity (9.8 meters

per second squared).

(a) If you can throw a baseball with an initial speed of 34.8 meters per second, at what angle of elevation $\theta$should you direct the throw so that the ball travels a distance of 107 meters before striking the ground?

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

(c) Graph $R=R\left(\theta \right),$with $v_0=34.8$meters per second.

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

If $\tan \alpha =x+1$and role="math" localid="1646425111199" $\tan \beta =x-1,$show that

role="math" localid="1646426027733" $2cot(\alpha -b)=x^2$