Question

timed problem
score: 8/10 current time: 330.6
find: cos a
the figure is not drawn to scale.
right triangle with legs 20, 21, hypotenuse 29, angle a at the vertex with leg 21
answer options: \\(\frac{20}{21}\\), \\(\frac{21}{20}\\), \\(\frac{20}{29}\\), \\(\frac{21}{29}\\)

Explanation:

Step1: Recall cosine definition

In a right triangle, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Identify sides for angle A

For $\angle A$, adjacent side is 21, hypotenuse is 29.

Step3: Calculate $\cos A$

$\cos A = \frac{21}{29}$? Wait, no, wait: Wait, the right triangle has legs 20 and 21, hypotenuse 29. Wait, angle A: let's check the triangle. The right angle is between the sides 20 and 21. So angle A is at the vertex with sides 21 (adjacent) and 20 (opposite), hypotenuse 29. Wait, no: adjacent to angle A is the leg adjacent, so adjacent is 21? Wait, no, wait: in a right triangle, for angle A, the adjacent side is the one that is part of angle A and not the hypotenuse. So angle A is at the end of the side 21 and hypotenuse 29. So the adjacent side to angle A is 21, opposite is 20, hypotenuse 29. Wait, but wait, cosine is adjacent over hypotenuse. Wait, but let's recheck. Wait, the triangle: right angle, so sides: one leg is 20 (horizontal), one leg is 21 (vertical), hypotenuse 29. Angle A is at the bottom left, so the sides: adjacent to angle A is the vertical leg (21), opposite is the horizontal leg (20), hypotenuse 29. So $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{21}{29}$? Wait, no, wait the options: one of the options is $\frac{21}{29}$? Wait the options are $\frac{20}{21}$, $\frac{21}{20}$, $\frac{20}{29}$, $\frac{21}{29}$. Wait, maybe I mixed up adjacent and opposite. Wait, angle A: let's label the triangle. Let's say the right angle is C, so triangle ABC, right-angled at C. Then side AC = 21, BC = 20, AB = 29. Then angle A is at vertex A, so the sides: adjacent to angle A is AC (21), opposite is BC (20), hypotenuse AB (29). So $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{21}{29}$. Wait, but let's check the options. The last option is $\frac{21}{29}$. Wait, but maybe I made a mistake. Wait, no, wait: maybe the adjacent side is 20? Wait, no, angle A: in triangle ABC, right-angled at C, angle at A: the sides forming angle A are AC and AB. So AC is adjacent, BC is opposite. So AC is 21, so adjacent is 21, hypotenuse 29. So $\cos A = \frac{21}{29}$. Wait, but let's check the options. The options are:

1. $\frac{20}{21}$

1. $\frac{21}{20}$

1. $\frac{20}{29}$

1. $\frac{21}{29}$

So the correct answer should be $\frac{21}{29}$. Wait, but let's re-express. Wait, maybe I mixed up the angle. Wait, angle B: let's see, angle B would have adjacent side 20, hypotenuse 29, so $\cos B = \frac{20}{29}$. But the question is $\cos A$. So angle A: adjacent is 21, hypotenuse 29, so $\cos A = \frac{21}{29}$. So the last option is $\frac{21}{29}$.

Answer:

$\frac{21}{29}$ (corresponding to the fourth option, $\frac{21}{29}$)