1) Use the limit definition of the derivative \(( \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } ) \) together with the tangent addition formula to prove that \(\frac { d } { d x } [ \tan x ] = \sec ^ { 2 } x\) 

Tangent Addition Formula: \(\tan ( x + y ) = \frac { \tan x + \tan y } { 1 - \tan x \tan y } \) 

You may also use this graph: \(f ( x ) = \frac { \tan x } { x } \) 

In triangle \(A B C\) , if altitude \(A D\) is drawn to side \(B C\) , which of the following must be true? 

(1.) \(\angle ADB \cong \angle A D C\) 

2. \(\overline { B D } \cong \overline { D C } \) 

3. \(\triangle A D B \cong \triangle A D C\) 

4. \(\angle B \cong \angle C\) 

Find the exact value of each expression, if it exists. 

\( \tan ( \sin ^ { - 1 } 1 - \cos ^ { - 1 } \frac { 1 } { 2 } ) \) 

a. \( \frac { \pi } { 3 } \) 

b. \( \frac { \sqrt { 3 } } { 3 } \) 

c. \( - \frac { \sqrt { 3 } } { 3 } \) 

d. undefined

Find all the solutions to \(\displaystyle \left ( { 2t - 3} \right ) \tan \left ( { \frac { { 6t} } { { 11} } } \right ) = 15 - 10t\). Use at least 4 decimal places in your work.

Find the exact value of each expression, if it exists. 

arcsin \( ( - \frac { 1 } { 2 } ) \) 

a. \( \frac { \pi } { 6 } \) 

b. \( - \frac { \pi } { 6 } \) 

c. \( - \frac { 1 } { 2 } \) 

d. undefined 

Find the exact value of the sine function of the given angle. 

\( 690 ^ { \circ } \) 

\( \sin 690 ^ { \circ } = \) 

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)