# SAT II Math I : Finding Sides with Trigonometry

## Example Questions

### Example Question #1 : Trigonometry

A plane flies  degrees north of east for  miles.  It then turns and flies  degrees south of east for  miles.  Approximately how many miles is the plane from its starting point?  (Ignore the curvature of the Earth.)

Explanation:

The plane flies two sides of a triangle.  The angle formed between the two sides is 40 degrees.  In a Side-Angle-Side situation, it is appropriate to employ the use of the Law of Cosines.

### Example Question #1 : Trigonometry

In :

Evaluate  to the nearest degree.

Insufficient information is provided to answer the question.

Explanation:

The figure referenced is below:

By the Law of Cosines, the relationship of the measure of an angle  of a triangle and the three side lengths , and  the sidelength opposite the aforementioned angle, is as follows:

All three sidelengths are known, so we are solving for . Setting

. the length of the side opposite the unknown angle;

;

;

and ,

We get the equation

Solving for :

Taking the inverse cosine:

,

the correct response.

### Example Question #1 : Trigonometry

In :

Evaluate the length of  to the nearest tenth of a unit.

Explanation:

The figure referenced is below:

By the Law of Cosines, given the lengths  and  of two sides of a triangle, and the measure  of their included angle, the length  of the third side can be calculated using the formula

Substituting , , and , then evaluating:

Taking the square root of both sides:

.

### Example Question #1 : Trigonometry

In :

Explanation:

The figure referenced is below:

The Law of Sines states that given two angles of a triangle with measures , and their opposite sides of lengths , respectively,

,

or, equivalently,

.

In this formula, we set:

, the desired sidelength;

, the measure of its opposite angle;

, the known sidelength;

, the measure of its opposite angle, which is

Substituting in the Law of Sines formula and solving for :

Evaluating the sines, then calculating: