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. 

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? 

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 \( 1 - 2 \cos ^ { 2 } ( x ) \) 

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.) 

\(\sin B = ?\)

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.)