Question 440·Easy·Math

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In right triangle $\triangle ABC$ , $\angle C$ is the right angle, $AC = 9$ , and $AB = 15$ . What is $\cos A$ ?

In a right triangle, a cosine ratio compares the leg next to the chosen angle with the hypotenuse, the side across from the right angle. Name those sides first, then divide their lengths and reduce the fraction. Don't switch to the opposite leg because it's another side of the triangle; that leg belongs in the sine ratio.

Hints

Hint 1
Find the hypotenuse first: it's the side across from the right angle, even though it also touches $A$ . Which given side lies across from $\angle C$ ?
Hint 2
For cosine, put the leg next to your chosen angle over the hypotenuse. Don't use the leg across from $A$ . Which of the two given lengths belongs on top?

Step-by-step

Name the sides, then use cosine
Step 1
Identify the sides from angle A
No Desmos needed. The two given sides are exactly the ones cosine compares. Since $\angle C$ is right, $AB=15$ is the hypotenuse, the side across from the right angle. Of the sides meeting at $A$ , $AC=9$ is the adjacent leg because it isn't the hypotenuse.
Step 2
Write the cosine ratio
For cosine, divide the adjacent leg by the hypotenuse. So $\cos A=\frac{AC}{AB}=\frac{9}{15}$ . Reversing the fraction would put the longer side on top, but cosine of an acute angle can't exceed $1$ .
Step 3
Reduce the fraction
Since $3$ divides both $9$ and $15$ , divide the numerator and denominator by the same number:
$\begin{aligned}\frac{9}{15}&=\frac{9\div3}{15\div3}\\[1.4em]&=\frac35.\end{aligned}$
So $\cos A$ is $\frac35$ . Choice D.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse sine, cosine, and tangent

Question 440·Easy·Math

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Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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