# SAT II Math I : Trigonometry

## Example Questions

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### 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 #2 : 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 #3 : 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 #4 : 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:

### Example Question #5 : Trigonometry

What is the measure of the angle made between a line segment with points  and the -axis?  Round your answer to the nearest hundreth of a degree.

No angle measure can be calculated

Explanation:

Based on the information given, we know that the ratio of  to  on this segment could be represented as:

This ratio represents the tangent of the triangle formed by our line segment and the -axis.  Using the inverse tangent function, we can find the angle measure:

This refers to a reference angle of

### Example Question #6 : Trigonometry

What is the measure of the angle made between a line segment with points  and the -axis? Round your answer to the nearest hundreth of a degree.

No angle can be calculated

Explanation:

Based on the information given, we know that the ratio of  to  on this segment could be represented as:

This ratio represents the tangent of the triangle formed by our line segment and the -axis. Using the inverse tangent function, we can find the angle measure:

This refers to a reference angle of .

### Example Question #1 : Trigonometry

A triangle is formed by connecting the points .  Determine the elevation angle to the nearest integer in degrees.

Explanation:

After connecting the points on the graph, the length of the triangular base is 1 unit.

The height of the triangle is 6.  To find the elevation angle, the angle is opposite from the height of the triangle.  Since we know the base and the height, the elevation angle can be solved by using the property of tangent.

### Example Question #8 : Trigonometry

Solve for between .

Explanation:

First we must solve for when sin is equal to 1/2. That is at

Now, plug it in:

### Example Question #1 : Trigonometry

Solve for between .

Explanation:

First we must solve for when sin is equal to 1/2. That is at

Now, plug it in:

### Example Question #10 : Trigonometry

In a triangle, , what is the measure of angle A if the side opposite of angle A is 3 and the adjacent side to angle A is 4?

(Round answer to the nearest tenth of a degree.)