All ACT Math Resources
Example Questions
Example Question #1 : How To Find Positive Cosine
The value of a cosine is positive in which quadrants?
The 1st and 4th
The 3rd only
The 1st and 3rd
The 4th only
The 1st and 4th
The cosine is positive in the 1st and 4th quadrants and negative in 2nd and 3rd
Example Question #1 : How To Find Positive Cosine
Which of the following is equal to
?
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 #2 : How To Find Positive Cosine
and is between and . What is the value of ?
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 #2 : How To Find Positive Cosine
To the nearest
, what is the cosine formed from the origin to ? Assume counterclockwise rotation.
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 #131 : Trigonometry
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 .
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
.