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Example Question #121 : Trigonometry

If $\tan A=\frac{b}{c}$ , where $b>0\ and\ c>0$ and $\frac{\pi }{2}<A<\pi$ ,

then what is $cos\ A$ ?

Possible Answers:

$\frac{c}{\sqrt{c^{2}+b^{2}}}$

$-\frac{\sqrt{c^{2}+b^{2}}}{c}$

$\frac{b}{\sqrt{c^{2}+b^{2}}}$

$\frac{\sqrt{c^{2}+b^{2}}}{c}$

$-\frac{c}{\sqrt{c^{2}+b^{2}}}$

Correct answer:

$-\frac{c}{\sqrt{c^{2}+b^{2}}}$

Explanation:

In the triangle below, the tangent of $\angle A\ is\ \frac{b}{c}$ , or the opposite side of the angle divided by the adjacent side of the angle. According to the Pythagorean

Theorem, the $hypotenuse^{2}=c^{2}+b^{2}$

Thus the hypotenuse equals $\sqrt{b^{2}+c^{2}}$ .

The cosine of an angle is the adjacent side of the angle divided by the hypotenuse of the triangle, giving us $\frac{c}{\sqrt{c^{2}+b^{2}}}$ .

However, since $tanA$ is $\frac{sinA}{cosA}$ , and when $A$ is between $\frac{\pi }{2}\ and\ \pi$ , $sinA$ is positive while $cosA$ is negative. Thus, $c$ is negative, giving us the final answer of $-\frac{c}{\sqrt{c^{2}+b^{2}}}$ . Triangle

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Example Question #121 : Trigonometry

What is cos θ?

Screen_shot_2013-07-15_at_9.54.23_pm

Possible Answers:

$\frac{\sqrt{29}}{29}$

$\frac{5}{29}$

$\frac{5\sqrt{29}}{29}$

$\frac{29}{5}$

Correct answer:

$\frac{5\sqrt{29}}{29}$

Explanation:

cos = adjacent/hypotenuse = $\frac{5}{\sqrt{29}}$

To get the radical out of the denominator, we multiply:

$\frac{\sqrt{29}}{\sqrt{29}}\ast\frac{5}{\sqrt{29}}$

$=\frac{5\sqrt{29}}{29}$

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Example Question #121 : Trigonometry

What is the cosine of the angle formed between the -axis and the line passing through (4,5) with a slope of 2.5 ? Round to the nearest hundredth.

Possible Answers:

0.4

0.34

0.6

0.91

0.14

Correct answer:

0.14

Explanation:

You do not even need to compute the line for this question. All that you need to do is note that you can make a little right triangle with a height of 2.5 and a base of , which you get from the slope of the line. So, to compute the cosine, you will need to find the hypotenuse using the Pythagorean theorem:

$h=\sqrt{2.5^2+1^2}=\sqrt{7.25}$

So, our little triangle looks like:

Cos1

The cosine will be the adjacent side, , divided by the hypotenuse, $\sqrt{7.25}$ :

$\frac{1}{\sqrt{7.25}}$ , or approximately 0.14 .

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Example Question #3 : How To Find The Cosine Of An Angle

What is the $\cos$ of an angle with $\sin \left ( \frac{4}{5} \right )$ ?

Possible Answers:

$\frac{3}{5}$

$\frac{5}{4}$

$\frac{3}{4}$

$\frac{5}{3}$

$\frac{4}{3}$

Correct answer:

$\frac{3}{5}$

Explanation:

Since the sine of the angle is $\frac{4}{5}$ , that means the opposite side of the triangle is and the hypotenuse is . Automatically, you know this is a special 3-4-5 right triangle, and that the missing side is 3. If not, you could also find the 3rd side by doing Pythagorean Theorem. This gives you your answer of $\cos=\frac{3}{5}$

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Example Question #4 : How To Find The Cosine Of An Angle

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

Possible Answers:

$\frac{5}{7}$

$\frac{5}{\sqrt74}$

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

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

$\frac{7}{5}$

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 #5 : How To Find The Cosine Of An Angle

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

Possible Answers:

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

$\frac{18}{7}$

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

$\frac{7}{18}$

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

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 3rd only

The 4th only

The 1st and 3rd

The 1st and 4th

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 #1 : How To Find Positive Cosine

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

Possible Answers:

$tan\ x$

$sec\ x$

$cot\ x$

$cot\ x* sec\ x$

$sin \ x* sec\ x$

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:

$3\sqrt{3}$

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

$\sqrt{2}$

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 #4 : 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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ACT Math : ACT Math

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #121 : Trigonometry

Example Question #121 : Trigonometry

Example Question #121 : Trigonometry

Example Question #3 : How To Find The Cosine Of An Angle

Example Question #4 : How To Find The Cosine Of An Angle

Example Question #5 : How To Find The Cosine Of An Angle

Example Question #1 : How To Find Positive Cosine

Example Question #1 : How To Find Positive Cosine

Example Question #2 : How To Find Positive Cosine

Example Question #4 : How To Find Positive Cosine

All ACT Math Resources

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