Math / Trigonometry

QuestionQuestion 14 of 25 Select the true statement about triangle $A B C$ . A. $\cos A=\cos C$ B. $\cos A=\sin B$ PREVIOUS

Studdy Solution

STEP 1

What is this asking? Which statement about the cosine and sine of the angles in this right triangle is true? Watch out! Don't mix up sine and cosine!
Remember SOH CAH TOA!

STEP 2

1. Find cos A
2. Find cos C
3. Find sin B
4. Check the statements

STEP 3

Alright, let's find $\cos A$ !
Remember CAH: Cosine is Adjacent over Hypotenuse.

STEP 4

Looking at angle $A$ , the adjacent side is $AB = 12$ and the hypotenuse is $AC = 13$ .

STEP 5

So, $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{12}{13}$ .
Awesome!

STEP 6

Now, let's find $\cos C$ !
Same idea, CAH: Cosine is Adjacent over Hypotenuse.

STEP 7

For angle $C$ , the adjacent side is $BC = 5$ and the hypotenuse is still $AC = 13$ .

STEP 8

Therefore, $\cos C = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{5}{13}$ .
Fantastic!

STEP 9

Time to find $\sin B$ !
Remember SOH: Sine is Opposite over Hypotenuse.

STEP 10

For angle $B$ , the opposite side is $AC = 13$ (that's the hypotenuse of the whole triangle!) and the hypotenuse is also $AC = 13$ because $B$ is the right angle.

STEP 11

So, $\sin B = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AC}{AC} = \frac{13}{13} = 1$ .
Perfect!

STEP 12

Is statement A true?
Is $\cos A = \cos C$ ?
We found $\cos A = \frac{12}{13}$ and $\cos C = \frac{5}{13}$ .
Nope, those are different!

STEP 13

Is statement B true?
Is $\cos A = \sin B$ ?
We found $\cos A = \frac{12}{13}$ and $\sin B = 1$ .
Still not the same!

STEP 14

Let's think about the other option, $\sin A$ .
Since $\sin A$ is Opposite over Hypotenuse, we get $\frac{5}{13}$ .
Hey, that's the same as $\cos C$ !
So, $\sin A = \cos C$ , which means $\sin A = \frac{5}{13}$ .

STEP 15

Also, since $\cos A = \frac{12}{13}$ , and $B$ is $90$ degrees, we know that $\sin(90 - A) = \cos A$ .
Since $A + B + C = 180$ and $B = 90$ , we know $A + C = 90$ , so $C = 90 - A$ .
Therefore, $\sin C = \frac{12}{13}$ .

STEP 16

The correct statement is $\sin A = \cos C$ or $\sin C = \cos A$ .

STEP 17

The true statement is that $\sin A = \cos C$ , even though that wasn't an option.
Neither of the provided statements A or B are true.

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