How to find positive cosine - ACT Math

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Question

The value of a cosine is positive in which quadrants?

Answer

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

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Question

Which of the following is equal to $cos\ x * csc\ x$ ?

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.

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Question

and is between and $\pi$ . What is the value of ?

Answer

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

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Question

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

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, $c = \sqrt{61}$

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

$\textup{cos }x^{\circ} = \frac{5}{\sqrt{61}} \approx 0.6402$

Thus, our cosine is approximately .

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Question

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 .

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, $c = \sqrt{436} = 2\sqrt{109}$

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

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

Thus, our cosine is approximately .

Compare your answer with the correct one above

Question

The value of a cosine is positive in which quadrants?

Answer

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

Compare your answer with the correct one above

Question

Which of the following is equal to $cos\ x * csc\ x$ ?

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.

Compare your answer with the correct one above

Question

and is between and $\pi$ . What is the value of ?

Answer

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

Compare your answer with the correct one above

Question

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

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, $c = \sqrt{61}$

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

$\textup{cos }x^{\circ} = \frac{5}{\sqrt{61}} \approx 0.6402$

Thus, our cosine is approximately .

Compare your answer with the correct one above

Question

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 .

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, $c = \sqrt{436} = 2\sqrt{109}$

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

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

Thus, our cosine is approximately .

Compare your answer with the correct one above

Question

The value of a cosine is positive in which quadrants?

Answer

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

Compare your answer with the correct one above

Question

Which of the following is equal to $cos\ x * csc\ x$ ?

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.

Compare your answer with the correct one above

Question

and is between and $\pi$ . What is the value of ?

Answer

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

Compare your answer with the correct one above

Question

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

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, $c = \sqrt{61}$

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

$\textup{cos }x^{\circ} = \frac{5}{\sqrt{61}} \approx 0.6402$

Thus, our cosine is approximately .

Compare your answer with the correct one above

Question

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 .

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, $c = \sqrt{436} = 2\sqrt{109}$

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

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

Thus, our cosine is approximately .

Compare your answer with the correct one above

Question

The value of a cosine is positive in which quadrants?

Answer

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

Compare your answer with the correct one above

Question

Which of the following is equal to $cos\ x * csc\ x$ ?

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.

Compare your answer with the correct one above

Question

and is between and $\pi$ . What is the value of ?

Answer

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

Compare your answer with the correct one above

Question

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

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, $c = \sqrt{61}$

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

$\textup{cos }x^{\circ} = \frac{5}{\sqrt{61}} \approx 0.6402$

Thus, our cosine is approximately .

Compare your answer with the correct one above

Question

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 .

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, $c = \sqrt{436} = 2\sqrt{109}$

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

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

Thus, our cosine is approximately .

Compare your answer with the correct one above

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