GOAL Calculate a resultant angle from two reflections.

PROBLEM Two mirrors make an angle of $120$ with each other, as in the figure. A ray is incident on mirror $M_1$ at an angle of $65$ to the normal. Find the angle the ray makes with the normal to $M_2$ after it is reflected Mirrors $M_1$ and $M_2$ make an angle of $120$ with each other. from both mirrors.

STRATEGY Apply the law of reflection twice. Given the incident ray at angle ${}_{\text{i}\text{n}\text{c}},$ find the final resultant angle, ${}_{\text{r}\text{e}\text{f}\text{f}}$

SOLUTION

Apply the law of reflection to $M_1$ to find the angle of reflection, ${}_{\text{r}\text{e}\text{f}}.$

${}_{\text{r}\text{e}\text{f}}={}_{\text{i}\text{n}\text{c}}=65$

Find the angle $$ that is the complement of the angle ${}_{\text{r}\text{e}\text{f}}$

$=90-{}_{\text{r}\text{e}\text{f}}=90-65=25$

Find the unknown angle $$ in the triangle of $M_1,M_2,$ and the ray

$180=25+120+=35$

traveling from $M_1$ to $M_2,$ using the fact $/$

that the three angles sum to $180.$

The angle $$ is complementary to the angle of incidence, ${}_{\text{i}\text{n}\text{c}},$ for $M_2.$

$+{}_{\text{i}\text{n}\text{c}}=90{}_{\text{i}\text{n}\text{c}}=90-35=55$

Apply the law of reflection a second time, obtaining ${}_{\text{r}\text{e}\text{f}}$

${}_{\text{r}\text{e}\text{f}}={}_{\text{i}\text{n}\text{c}}=55$

LEARN MORE

REMARKS Notice the heavy reliance on elementary geometry and trigonometry in these reflection problems.

QUESTION In general, what is the relationship between the incident angle ${}_{\text{i}\text{n}\text{c}}$ and the final reflected angle ${}_{\text{r}\text{e}\text{f}}$ when the angle between the mirrors is $90\mathrm{.}0?$

${}_{\text{i}\text{n}\text{c}}-{}_{\text{r}\text{e}\text{f}}=90\mathrm{.}0$

${}_{\text{i}\text{n}\text{c}}+{}_{\text{r}\text{e}\text{f}}=180\mathrm{.}0$

${}_{\text{i}\text{n}\text{c}}+{}_{\text{r}\text{e}\text{f}}=90\mathrm{.}0$

PRACTICE IT

Use the worked example above to help you solve this problem. Two mirrors make an angle of $119$ with each other, as in the figure. A ray is incident on mirror $M_1$ at an angle of ${}_{\text{i}\text{n}\text{c}}=58$ to the normal. Find the angle the ray makes with the normal to $M_2$ after it is reflected from both mirrors.

EXERCISE

HINTS: GETTING STARTED I I$\text{'}$M STUCK$$

Repeat the problem if the angle of incidence is $67$ and the second mirror makes an angle of $99$ with the first mirror. $=$