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ACT Math : Trigonometry

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Example Questions

Example Question #3041 : Act Math

Right triangle $\Delta ABC$ has sides AB=5 , BC = 7 and $AC = \sqrt{74}$ . What is the cosine of $\angle A$ ?

Possible Answers:

$\frac{5}{7}$

$\frac{\sqrt{74}}{5}$

$\frac{5}{\sqrt74}$

$\frac{7}{5}$

$\frac{7}{\sqrt{74}}$

Correct answer:

$\frac{5}{\sqrt74}$

Explanation:

SOHCAHTOA tells us that $\textup{cos} = \frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know the hypotenuse is the longest side of the triangle, $\sqrt(74)$ . Our adjacent side will be the other side that has as a vertex, AB = 5 .

Thus, $\textup{cos} (A) = \frac{AB}{AC} = \frac{5}{\sqrt{74}}$ .

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Example Question #3041 : Act Math

Right triangle $\Delta ABC$ has sides AB=7 , $BC = 5\sqrt{11}$ and AC = 18 . What is the cosine of $\angle A$ ?

Possible Answers:

$\frac{5\sqrt{11}}{18}$

$\frac{7}{5\sqrt{11}}$

$\frac{18}{7}$

$\frac{7}{18}$

$\frac{5\sqrt{11}}{7}$

Correct answer:

$\frac{7}{18}$

Explanation:

SOHCAHTOA tells us that $\textup{cos} = \frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know the hypotenuse is the longest side of the triangle, . Our adjacent side will be the other side that has as a vertex, AB = 7 .

Thus, $\textup{cos} (A) = \frac{AB}{AC} = \frac{7}{18}$ .

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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 1^st and 4^th quadrants and negative in 2^nd and 3^rd

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Example Question #32 : Cosine

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

Possible Answers:

$sin \ x* sec\ x$

$cot\ x* sec\ x$

$tan\ x$

$cot\ x$

$sec\ x$

Correct answer:

$cot\ x$

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.

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Example Question #2 : How To Find Positive Cosine

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

Possible Answers:

$\sqrt{2}$

$\frac{\sqrt{3}}{2}$

$3\sqrt{3}$

0.25

Correct answer:

$\frac{\sqrt{3}}{2}$

Explanation:

For to $\pi$ , we know that $cos(\frac{\pi}{3}) = 0.5$ . So, the question asks, what is the value of cos(0.5x) , 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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Example Question #2 : How To Find Positive Cosine

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

Possible Answers:

Correct answer:

Explanation:

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

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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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 .

Possible Answers:

Correct answer:

Explanation:

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 a^2 + b^2 = c^2 , so we plug in and find that: 20^2 + 6^2 = c^2 = 436

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 .

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Example Question #1 : How To Find The Period Of The Cosine

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$cos\, (2x)$ .

Possible Answers:

$\pi+2$

$\pi$

$2\pi$

Correct answer:

$\pi$

Explanation:

For the function

cos(Ax)

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{2}$

which reduces to $\pi$ .

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Example Question #131 : Trigonometry

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$\cos \bigg(\frac{1}{4}\pi x\bigg)$ .

Possible Answers:

$4\pi$

$8\pi$

Correct answer:

Explanation:

For the function

cos(Ax)

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{\frac{1}{4}\pi}=2\pi \cdot \frac{4}{1\pi}$

which reduces to .

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Example Question #3051 : Act Math

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$cos\, (\pi x)$ .

Possible Answers:

$\pi$

$2\pi$

Correct answer:

Explanation:

For the function

cos(Ax)

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{\pi}$

which reduces to .

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ACT Math : Trigonometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #3041 : Act Math

Example Question #3041 : Act Math

Example Question #1 : How To Find Positive Cosine

Example Question #32 : Cosine

Example Question #2 : How To Find Positive Cosine

Example Question #2 : How To Find Positive Cosine

Example Question #131 : Trigonometry

Example Question #1 : How To Find The Period Of The Cosine

Example Question #131 : Trigonometry

Example Question #3051 : Act Math

All ACT Math Resources

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