91Ó°ÊÓ

The sine of an angle between two sides of a triangle: \(\sin \theta=\frac{2 A}{a b}\) If the area \(A\) and two sides \(a\) and \(b\) of a triangle are known, the sine of the angle between the two sides is given by the formula shown. Find the angle \(\theta\) for the triangle shown given \(A=38.9\), and use it to solve the triangle. (Hint: Apply the same concept to angle \(\gamma\) or \(\beta\).)

Illumination of a surface: \(E=\frac{I \cos \theta}{d^{2}}\) The illumination \(E\) of a surface by a light source is a measure of the luminous flux per unit area that reaches the surface. The value of \(E\) [in lumens (lm) per square foot] is given by the formula shown, where \(d\) is the distance from the light source (in feet), \(I\) is the intensity of the light [in candelas (cd)], and \(\theta\) is the angle the light source makes with the vertical. For reading a book, an illumination \(E\) of at least \(18 \mathrm{~lm} / \mathrm{ft}^{2}\) is recommended. Assuming the open book is lying on a horizontal surface, how far away should a light source be placed if it has an intensity of \(90 \mathrm{~cd}\) (about \(75 \mathrm{~W}\) ) and the light flux makes an \text { angle of } 65^{\circ} \text { with the book's surface (i.e., } \theta=25^{\circ} \text { )? }

For each exercise, state the quadrant of the terminal side of \(\theta\) and the slgn of the function in that quadrant. Then find the reference angle \(\theta_{r}\) and evaluate the function at both \(\theta\) and \(\theta_{r}\) using a calculator. Round to four decimal places. \(\theta=772^{\circ}\); sine

The area of a parallelogram: \(A=a b \sin \theta\) The area of a parallelogram is given by the formula shown, where \(a\) and \(b\) are the lengths of the sides and \(\theta\) is the angle between them. Use the formula to complete the following: (a) find the area of a parallelogram with sides \(a=9\) and \(b=21\) given \(\theta=50^{\circ}\). (b) What is the smallest integer value of \(\theta\) where the area is greater than 150 units \({ }^{2}\) ? (c) State what happens when \(\theta=90^{\circ}\). (d) How can you find the area of a triangle using this formula?

The angle between two intersecting lines: $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{2} m_{1}} $$ Given line 1 and line 2 with slopes \(m_{1}\) and \(m_{2}\), respectively, the angle between the two lines is given by the formula shown. Find the angle \(\theta\) if the equation of line 1 is \(y_{1}=\frac{3}{4} x+2\) and line 2 has equation \(y_{2}=-\frac{2}{3} x+5\).

Diagonal of a rectangular parallelepiped: A rectangular box has a width of \(25 \mathrm{~cm}\), a length of \(45 \mathrm{~cm}\), and a height of \(10 \mathrm{~cm}\). Find (a) the length of the diagonal that passes through the center of the box and (b) the cosine of the angle \(\theta\) it makes at the lower comer of the box.

Find all angles satisfying the stated relationship. For standard angles, express your answer in exact form. For nonstandard values, use a calculator and round function values to tenths. $$\tan \theta=-\frac{\sqrt{3}}{1}$$

Nonacute angles: At a recent carnival, one of the games on the midway was playcd using a large spinner that turns clockwise. On Jorge's spin the number 25 began at the 12 o'clock (top/center) position, returned to this position five times during the spin and stopped at the 3 o'clock position. What angle 8 did the spinner spin through? Name all angles that are coterminal with \(\theta\).

In an elementary study of trigonometry, the hands of a clock are often studied because of the angle relationship that exists between the hands. For example, at 3 o'clock, the angle between the two hands is a right angle and measures \(90^{\circ}\). a. What is the angle between the two hands at 1 o'clock? \(20^{\prime}\) clock? Explain why. b. What is the angle between the two hands at \(6: 30 ? 7: 00 ? 7: 30\) ? Explain why. c. Name four times at which the hands will form a \(45^{\circ}\) angle.