# Tutor profile: Will B.

## Questions

### Subject: Trigonometry

A triangle has one side with angle 45 degrees. The two adjacent sides have lengths of 2 and 4. Find the length of the third side.

We can solve this with the law of cosines, $$c=\sqrt{a^2+b^2-2abcos(\gamma)}$$, where a and b are the adjacent sides, c is the opposite side, and $$\gamma$$ is the angle. Plugging into this equation a=2, b=4, and $$\gamma$$=45 degrees, we find $$c=\sqrt{2^2+4^2-2*4cos(45)}=\sqrt{4+16-\frac{8}{\sqrt{2}}}$$, or approximately 5.07.

### Subject: Calculus

Solve the indefinite integral $$\int sin(x)cos(x)dx$$.

This problem can be simplified with u substitution. Let $$u=sin(x)$$. Then $$du=cos(x)dx$$. We can substitute this into the equation to get $$\int u du$$. Using the power rule, we simply fund that this integral is equal to $$\frac{1}{2} u^2+C$$ for some constant C. Plugging back in $$u=sin(x)$$, we find that the integral is equal to $$\frac{1}{2} sin(x)^2+C$$. Using the chain rule to differentiate this confirms that we have performed the integral correctly.

### Subject: Physics

An stationary object explodes into three pieces of shrapnel of equal mass, A, B, and C. A travels directly east, B travels directly south, and C travels at 3 m/s at 30 degrees north of west. What are the velocities of A and B?

By conservation of momentum, we can break up the vertical and horizontal momentum of C to find the momentum (and therefore velocity) of A and B. All shrapnels have the same mass, so it cancels. A's momentum is equal to C's horizontal momentum, so the velocity of A is $$3 cos(30)$$ east. Similarly, B's velocity is $$3sin(30)$$ south.