Trigonometry #2

We know that, $\cos { 45 } =\frac { 1 }{ \sqrt { 2 } } \quad or\quad \cos { 315 } =\frac { 1 }{ \sqrt { 2 } }$ $\Rightarrow \frac { 1 }{ 2 } \theta +37=45\quad or\quad \frac { 1 }{ 2 } \theta +37=315$ Solving both the equations for $\theta$ , we get $\boxed { \theta =16 }$ as the only value such that $0<\theta <360$

Since it is $cos(\frac { 1 }{ 2 } \theta +37°)$ , the graph is $2$ times longer than $cos(x)$ and shifted $37°$ to the left.

The possible values of $x$ for $cos(x)=\frac { 1 }{ \sqrt { 2 } }$ is $45°$ and $315°$ if $x$ is between $0°$ and $360°$

Shifting $cos(x)$ to the left by $37°$ would give a graph of $cos(x+37°)$ , which the solution for $cos(x+37°)=\frac { 1 }{ \sqrt { 2 } }$ is $45°-37°=8°$ and $315°-37°=278°$ .

Now, making it 2 times longer would give the graph of $cos(\frac { 1 }{ 2 } x+37°)$ , so the solution for this is $8°*2=16°$ therefore, the answer is $\boxed { 16 }$

Note that $278°*2=556°$ is not counted as it is more than $360°$

The graph is now $\times2$ longer than before so we have to minus $360\times2$ instead of $360$ , which would give a negative value and thus isn't counted.