91Ó°ÊÓ

Explain the mistake that is made. Use the unit circle to evaluate \(\tan \left(\frac{5 \pi}{6}\right)\) exactly. \(\begin{array}{l}\text { Tangent is the } \\ \text { ratio of sine } \\\ \text { to cosine. }\end{array} \tan \left(\frac{5 \pi}{6}\right)=\frac{\sin \left(\frac{5 \pi}{6}\right)}{\cos \left(\frac{5 \pi}{6}\right)}\) Use the unit circle to \(\begin{array}{l}\text { identify sine } \\ \text { and cosine. }\end{array} \quad \sin \left(\frac{5 \pi}{6}\right)=-\frac{\sqrt{3}}{2}\) and \(\cos \left(\frac{5 \pi}{6}\right)=\frac{1}{2}\) Substitute values for \(\begin{array}{l}\text { sine and } \\ \text { cosine. }\end{array} \quad \tan \left(\frac{5 \pi}{6}\right)=\frac{-(\sqrt{3} / 2)}{1 / 2}\) Simplify. \(\tan \left(\frac{5 \pi}{6}\right)=-\sqrt{3}\) This is incorrect. What mistake was made?

In Exercises \(61-66,\) sketch the graph of the function over the indicated interval. $$y=4-3 \cos [\pi(x+1)],[-1,3]$$

Explain the mistake that is made. Use the unit circle to evaluate \(\sec \left(\frac{11 \pi}{6}\right)\) exactly. Solution: \(\begin{array}{l}\text { Secant is the reciprocal } \\ \text { of cosine. }\end{array} \quad \sec \left(\frac{11 \pi}{6}\right)=\frac{1}{\cos \left(\frac{11 \pi}{6}\right)}\) \(\begin{array}{l}\text { Use the unit circle } \\ \text { to evaluate cosine. }\end{array} \quad \cos \left(\frac{11 \pi}{6}\right)=-\frac{1}{2}\) \(\begin{array}{l}\text { Substitute the value } \\ \text { for cosine. }\end{array} \quad \sec \left(\frac{11 \pi}{6}\right)=\frac{1}{-\frac{1}{2}}\) Simplify. \(\quad \sec \left(\frac{11 \pi}{6}\right)=-2\) This is incorrect. What mistake was made?

In Exercises \(57-66,\) state the domain and range of the functions. $$y=\frac{1}{2}-\frac{1}{3} \csc \left(3 x-\frac{\pi}{2}\right)$$

Determine whether each statement is true or false. \(\sin (2 n \pi+\theta)=\sin \theta, n\) an integer.

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=2 x-\cos (\pi x), 0 \leq x \leq 4$$

Determine whether each statement is true or false. \(\cos (2 n \pi+\theta)=\cos \theta, n\) an integer.

A lighthouse is located on a small island 3 miles offshore. The distance \(x\) is given by \(x=3 \tan (\pi t)\) where \(t\) is the time measured in seconds. Suppose that at midnight the light beam forms a straight angle with the shoreline. Find \(x\) at a. \(t=\frac{2}{3} s\) b. \(t=\frac{3}{4} s\) c. \(1 \mathrm{s}\) d. \(t=\frac{5}{4} s\) e. \(t=\frac{4}{3} \mathrm{s}\) Round to the nearest length. PICTURE CANT COPY

Determine whether each statement is true or false. \(\sin \theta=1\) when \(\theta=\frac{(2 n+1) \pi}{2}, n\) an integer.

If the length of the light beam is determined by \(y=3|\sec (\pi t)|,\) find \(y\) at a. \(t=\frac{2}{3} s\) b. \(t=\frac{3}{4} s\) c. \(1 \mathrm{s}\) d. \(t=\frac{5}{4} \mathrm{s}\) e. \(t=\frac{4}{3} \mathrm{s}\) Round to the nearest length.