a. Use the binomial series and the fact that \(\frac { d } { d x } \sin ^ { - 1 } x = ( 1 - x ^ { 2 } ) ^ { - 1 / 2 }\)
Question
Answer
\(\left. \begin{array} { l } { \text { (a) } x + \frac { x ^ { 3 } } { 6 } + \frac { 3 x ^ { 5 } } { 40 } + \frac { 5 x ^ { 7 } } { 112 } , \text { radius of convergence } = 1 } \\ { \text { (b) } \frac { \pi } { 2 } - x - \frac { x ^ { 3 } } { 6 } - \frac { 3 x ^ { 5 } } { 40 } - \frac { 5 x ^ { 7 } } { 112 } } \end{array} \right.\)
Related Questions
Q:
What is the distance from Kareem to the house?
Kareem and Amy are standing on a riverbank, \( 150 \) meters apart, at points \( A \) and \( B \) respectively. (See the figure below.) They want to know the distance from Kareem to a house located across the river at point \( C \) . Kareem measures angle \( A \) (angle \( B A C \) ) to be \( 55 ^ { \circ } \) , and Amy measures angle \( B \) (angle \( A B C \) ) to be \( 74 ^ { \circ } \) . What is the distance from Kareem to the house? Round your answer to the nearest tenth of a meter.
Q:
Suppose a projectile is fired from a cannon with velocity \( v _ { 0 } \) and angle of elevation \( \theta \) . The horizontal distance \( R ( \theta ) \) it travels (in feet) is given by the following.
\( R ( \theta ) = \frac { ( v _ { 0 } ) ^ { 2 } \sin 2 \theta } { 32 } \)
If \( v _ { 0 } = 80 ft \) /s, what angle \( 0 \) (in radians) should be used to hit a target on the ground \( 118 \) feet in front of the cannon?
Q:
Consider the derivation of an alternate form of the cosine double angle identity.
1. \( \cos ( 2 x ) = \cos ^ { 2 } ( x ) - \sin ^ { 2 } ( x ) \)
2. \(= \cos ^ { 2 } ( x ) - ( 1 - \cos ^ { 2 } ( x ) ) \)
3. \(= \cos ^ { 2 } ( x ) - 1 - \cos ^ { 2 } ( x ) \)
4. \(= 2 \cos ^ { 2 } ( x ) - 1\)
What is the error in this derivation?
In step \( 1 , \cos ( 2 x ) \) is equal to \( \cos ^ { 2 } ( x ) + \sin ^ { 2 } ( x ) \) .
In step \( 2 , \sin ^ { 2 } ( x ) \) should have been replaced with \( 1 + \) \( \cos ^ { 2 } ( x ) \) .
In step \( 3 , \cos ^ { 2 } ( x ) - 1 - \cos ^ { 2 } ( x ) \) should be \( \cos ^ { 2 } ( x ) - 1 \) \( + \cos ^ { 2 } ( x ) \)
In \( \operatorname { step } 4 , 2 \cos ^ { 2 } ( x ) - 1 \) should be \( 1 - 2 \cos ^ { 2 } ( x ) \)
Q:
Suppose \( A B C \) is a right triangle with sides a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown s the denominators when apolicable. nide length. Then find the values of the six trigonometric functions for angle B. Rationalize the denominators when applicable. \(a = 3 , b = 4\)
What is the length of side c? (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
\(\sin B = ?\)
Q:
Write the ratios for \( \sin M , \cos M , \) and \( \tan M \) . Give the exact value and a four-decimal approximation.
\(\sin M = \square \)
(Type an exact answer in simplified form. Type an integer or a fraction.) Type the decimal approximation of the answer rounded to four decimal places.
\( \sin M = \square \)
(Round the final answer to four decimal places as needed.)