### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find Positive Cosine

The value of a cosine is positive in which quadrants?

**Possible Answers:**

The 3rd only

The 1st and 3rd

The 4th only

The 1st and 4th

**Correct answer:**

The 1st and 4th

The cosine is positive in the 1^{st} and 4^{th} quadrants and negative in 2^{nd} and 3^{rd}

### Example Question #1 : How To Find Positive Cosine

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

Here, we use the SOHCAHTOA ratios and the fact that csc x = 1 / sin x.

Cosine x = adjacent side length / hypotenuse length

Cosecant x = 1 / sin x = hypotenuse / opposite

(Adjacent / hypotenuse) * (hypotenuse / opposite) = Adjacent / opposite = Cotangent x.

### Example Question #1 : How To Find Positive Cosine

and is between and . What is the value of ?

**Possible Answers:**

**Correct answer:**

For to , we know that . So, the question asks, what is the value of , where . Therefore, it is asking what the value of is, which is .

### Example Question #3 : How To Find Positive Cosine

To the nearest , what is the cosine formed from the origin to ? Assume counterclockwise rotation.

**Possible Answers:**

**Correct answer:**

If the point to be reached is , then we may envision a right triangle with sides and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that:

Thus,

Now, **SOHCAHTOA** tells us that , so we know that:

Thus, our cosine is approximately .

### Example Question #4 : How To Find Positive Cosine

Two drivers race to a finish line. Driver A drives north blocks, and east blocks and crosses the goal. Driver B drives the shortest direct route between the two points. Relative to east, what is the cosine of the angle at which Driver B raced? Round to the nearest .

**Possible Answers:**

**Correct answer:**

If the point to be reached is blocks north and blocks east, then we may envision a right triangle with sides and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that:

Thus,

Now, **SOHCAHTOA** tells us that , so we know that:

Thus, our cosine is approximately .