Trigonometric Substitution In Exercises \(53 - 56\) , use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta\) , where
\(0 < \theta < \pi / 2\) . 53. \(\sqrt { 9 - x ^ { 2 } } , x = 3 \cos \theta\)
55. \(\sqrt { x ^ { 2 } - 4 } , x = 2 \sec \theta\)
53. \(3 \sqrt { ( 1 + \cos ( \theta ) ) ( 1 - \cos ( \theta ) ) }\)
55. \(2 \sqrt { ( \sec ( \theta ) + 1 ) ( \sec ( \theta ) - 1 ) }\)
Lupe and Byron start riding their bikes from the same location. Lupe rides \( 24 \) miles north while Byron rides \( 7 \) miles east. How far apart are Lupe and Byron when they stop riding? HINT: draw a picture.
Use a calculator to show that each of the following pairs of coterminal angles \( \theta \) and \( \beta \) have the same values, correct to four decimals, for the sine ratio.
a. \( \theta = 100 ^ { \circ } , \beta = 460 ^ { \circ } \) b. \( \theta = 245 ^ { \circ } , \beta = - 115 ^ { \circ } \)
c. \( \theta = - 75 ^ { \circ } , \beta = - 1155 ^ { \circ } \) d. \( \theta = 2000 ^ { \circ } , \beta = 920 ^ { \circ }\)
