ACT Math : How to find the cosine of an angle

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #30 : Cosine

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

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

\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\ \pisinA is positive while cosA is negative. Thus, c is negative, giving us the final answer of -\frac{c}{\sqrt{c^{2}+b^{2}}}.Triangle

Example Question #1 : How To Find The Cosine Of An Angle

What is cos θ?

Screen_shot_2013-07-15_at_9.54.23_pm

Possible Answers:

Correct answer:

Explanation:

cos = adjacent/hypotenuse = 

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

Example Question #2 : How To Find The Cosine Of An Angle

What is the cosine of the angle formed between the -axis and the line passing through  with a slope of ?  Round to the nearest hundredth.

Possible Answers:

Correct answer:

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  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:

So, our little triangle looks like:

Cos1

The cosine will be the adjacent side, , divided by the hypotenuse, :

, or approximately .

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

What is the  of an angle with ?

Possible Answers:

Correct answer:

Explanation:

Since the sine of the angle is , 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 

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

Right triangle  has sides  and . What is the cosine of 

Possible Answers:

Correct answer:

Explanation:

SOHCAHTOA tells us that , 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, .

Thus, .

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

Right triangle  has sides  and . What is the cosine of 

Possible Answers:

Correct answer:

Explanation:

SOHCAHTOA tells us that , 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, .

Thus, .

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