ACT Math : How to find positive cosine

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Find Positive Cosine

The value of a cosine is positive in which quadrants?

 

Possible Answers:

The 1st and 4th

The 3rd only

The 1st and 3rd

The 4th only

Correct answer:

The 1st and 4th

Explanation:

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 \(\displaystyle cos\ x * csc\ x\)?

Possible Answers:

\(\displaystyle cot\ x* sec\ x\)

\(\displaystyle tan\ x\)

\(\displaystyle sec\ x\)

\(\displaystyle cot\ x\)

\(\displaystyle sin \ x* sec\ x\)

Correct answer:

\(\displaystyle cot\ x\)

Explanation:

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

\(\displaystyle cos(x)=0.5\) and \(\displaystyle x\) is between \(\displaystyle 0\) and \(\displaystyle \pi\).  What is the value of \(\displaystyle cos(0.5x)\)?

Possible Answers:

\(\displaystyle \sqrt{2}\)

\(\displaystyle 1\)

\(\displaystyle \frac{\sqrt{3}}{2}\)

\(\displaystyle 3\sqrt{3}\)

\(\displaystyle 0.25\)

Correct answer:

\(\displaystyle \frac{\sqrt{3}}{2}\)

Explanation:

For \(\displaystyle 0\) to \(\displaystyle \pi\), we know that \(\displaystyle cos(\frac{\pi}{3}) = 0.5\).  So, the question asks, what is the value of \(\displaystyle cos(0.5x)\), where \(\displaystyle x=\frac{\pi}{3}\).  Therefore, it is asking what the value of \(\displaystyle cos(\frac{\pi}{6})\) is, which is \(\displaystyle \frac{\sqrt{3}}{2}\).

Example Question #2 : How To Find Positive Cosine

To the nearest \(\displaystyle .001\), what is the cosine formed from the origin to \(\displaystyle (5, 6)\)? Assume counterclockwise rotation.

Possible Answers:

\(\displaystyle .355\)

\(\displaystyle .710\)

\(\displaystyle .640\)

\(\displaystyle .333\)

\(\displaystyle .925\)

Correct answer:

\(\displaystyle .640\)

Explanation:

If the point to be reached is \(\displaystyle (5, 6)\), then we may envision a right triangle with sides \(\displaystyle 5\) and \(\displaystyle 6\), and hypotenuse \(\displaystyle c\). The Pythagorean Theorem tells us that \(\displaystyle a^2 + b^2 = c^2\), so we plug in and find that: \(\displaystyle 5^2 + 6^2 = c^2 = 61\)

Thus, \(\displaystyle c = \sqrt{61}\)

Now, SOHCAHTOA tells us that \(\displaystyle \textup{cosine} = \frac{\textup{adjacent}}{\textup{hypotenuse}}\), so we know that:

\(\displaystyle \textup{cos }x^{\circ} = \frac{5}{\sqrt{61}} \approx 0.6402\)

Thus, our cosine is approximately \(\displaystyle .640\).

Example Question #131 : Trigonometry

Two drivers race to a finish line. Driver A drives north \(\displaystyle 6\) blocks, and east \(\displaystyle 20\) 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 \(\displaystyle .001\).

Possible Answers:

\(\displaystyle .073\)

\(\displaystyle .700\)

\(\displaystyle .445\)

\(\displaystyle .836\)

\(\displaystyle .958\)

Correct answer:

\(\displaystyle .958\)

Explanation:

If the point to be reached is \(\displaystyle 6\) blocks north and \(\displaystyle 20\) blocks east, then we may envision a right triangle with sides \(\displaystyle 20\) and \(\displaystyle 6\), and hypotenuse \(\displaystyle c\). The Pythagorean Theorem tells us that \(\displaystyle a^2 + b^2 = c^2\), so we plug in and find that: \(\displaystyle 20^2 + 6^2 = c^2 = 436\)

Thus, \(\displaystyle c = \sqrt{436} = 2\sqrt{109}\)

Now, SOHCAHTOA tells us that \(\displaystyle \textup{cosine} = \frac{\textup{adjacent}}{\textup{hypotenuse}}\), so we know that:

\(\displaystyle cos x^{\circ} = \frac{20}{2\sqrt{109}} = \frac{10}{\sqrt{109}} \approx .9578\)

Thus, our cosine is approximately \(\displaystyle .958\).

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