Question

consider the work shown to solve for x in the following right triangle.
image of right triangle with angle 67.4°, hypotenuse 26, and leg x
\\(\cos(67.4^\circ) = \frac{26}{x}\\)
\\(x \cos(67.4^\circ) = 26\\)
\\(x = \frac{26}{\cos(67.4^\circ)}
which statement best describes the error in the work above?
multiple choice options, one marked as you did not select an option
correctly solve for x. round your answer to two decimal places. if there was no error, enter the value for x from the work above. enter deg after any degree value.

Explanation:

Step1: Identify the correct trigonometric ratio

In a right triangle, cosine of an angle is defined as the adjacent side over the hypotenuse. The angle given is \(67.4^\circ\), the side adjacent to this angle is \(x\), and the hypotenuse is \(26\)? Wait, no—wait, looking at the triangle, the right angle is at the bottom left, so the angle \(67.4^\circ\) is at the top left. So the side adjacent to \(67.4^\circ\) is \(x\), and the hypotenuse is \(26\)? Wait, no, actually, let's re-examine. Wait, the hypotenuse is the side opposite the right angle, so the hypotenuse is \(26\)? Wait, no, in the triangle, the side labeled \(26\) is the hypotenuse? Wait, no, the right angle is at the bottom left, so the two legs are \(x\) (vertical leg) and the horizontal leg, and the hypotenuse is the side opposite the right angle, which is the side labeled \(26\)? Wait, no, the angle at the top left is \(67.4^\circ\), so the adjacent side to \(67.4^\circ\) is \(x\) (vertical leg), and the hypotenuse is \(26\)? Wait, no, cosine of an angle is adjacent over hypotenuse. Wait, maybe the initial setup was wrong. Wait, actually, in the triangle, the angle \(67.4^\circ\), the adjacent side to it is \(x\), and the hypotenuse is \(26\)? Wait, no, maybe the opposite side? Wait, no, let's recall: in a right triangle, for angle \(\theta\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So the angle \(67.4^\circ\): the adjacent side is \(x\) (the vertical leg), the hypotenuse is \(26\)? Wait, no, the hypotenuse is the side opposite the right angle, so the hypotenuse is the side labeled \(26\). Wait, but then the adjacent side to \(67.4^\circ\) is \(x\), and the hypotenuse is \(26\), so \(\cos(67.4^\circ)=\frac{x}{26}\), not \(\frac{26}{x}\). Ah, so the initial setup was wrong. The correct equation should be \(\cos(67.4^\circ)=\frac{x}{26}\), not \(\frac{26}{x}\). So the error was in the initial trigonometric ratio setup. So to solve for \(x\), we should use \(\cos(67.4^\circ)=\frac{x}{26}\), so \(x = 26\times\cos(67.4^\circ)\).

Step2: Calculate the value of \(x\)

First, find \(\cos(67.4^\circ)\). Using a calculator, make sure it's in degree mode. \(\cos(67.4^\circ)\approx\cos(67.4)\approx0.3863\) (wait, no, wait, 67.4 degrees: let's calculate it properly. Wait, 67.4 degrees. Let's compute \(\cos(67.4^\circ)\). Using a calculator: \(\cos(67.4) \approx 0.386287\). Then \(x = 26\times0.386287\approx10.04\)? Wait, no, wait, maybe I mixed up adjacent and opposite. Wait, wait, the angle is at the top left, the right angle is at the bottom left, so the sides: the vertical leg is \(x\), the horizontal leg is the other leg, and the hypotenuse is 26. So the angle at the top left: the adjacent side is \(x\) (vertical), the opposite side is the horizontal leg. Wait, no, adjacent is the side next to the angle, so if the angle is at the top left, the sides forming the angle are \(x\) (vertical) and the hypotenuse? No, the two sides forming the angle \(67.4^\circ\) are the vertical leg (\(x\)) and the hypotenuse? No, the right angle is at the bottom left, so the triangle has vertices: top left (angle \(67.4^\circ\)), bottom left (right angle), and bottom right (the other acute angle). So the sides: bottom left to top left is \(x\) (vertical leg), bottom left to bottom right is horizontal leg, top left to bottom right is hypotenuse (26). So the angle at top left: the adjacent side is \(x\) (vertical leg), the hypotenuse is 26, and the opposite side is the horizontal leg. Therefore, \(\cos(67.4^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{x}{26}\), so the initial equation \(\cos(67.4^\circ)=\frac{26}{x}\) is wrong. The correct equation is \(\cos(67.4^\circ)=\frac{x}{26}\), so solving for \(x\), we get \(x = 26\times\cos(67.4^\circ)\). Wait, but let's check with sine. Wait, maybe the angle's opposite side is \(x\)? Wait, no, the angle at top left: the side opposite to it would be the horizontal leg, and the adjacent is \(x\). Wait, maybe I made a mistake in identifying adjacent and hypotenuse. Wait, let's use sine. Wait, \(\sin(67.4^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to \(67.4^\circ\) is the horizontal leg, and the adjacent is \(x\). Wait, no, maybe the initial problem had the hypotenuse as 26, and \(x\) is a leg. Wait, let's re-express. Let's denote: angle \(A = 67.4^\circ\), right angle at \(C\), so side \(AC = x\) (leg), side \(AB = 26\) (hypotenuse), side \(BC\) (leg). Then \(\cos(A)=\frac{AC}{AB}=\frac{x}{26}\), so \(x = 26\cos(A)\). So the initial setup was wrong; it should be \(\cos(67.4^\circ)=\frac{x}{26}\), not \(\frac{26}{x}\). So to solve for \(x\) correctly, we use \(x = 26\times\cos(67.4^\circ)\). Let's calculate that. First, compute \(\cos(67.4^\circ)\). Using a calculator (in degree mode): \(\cos(67.4) \approx 0.386287\). Then \(x = 26\times0.386287 \approx 10.04\)? Wait, no, that seems small. Wait, maybe the angle is such that \(x\) is the opposite side? Wait, maybe I mixed up the angle. Wait, if the right angle is at the bottom left, and the angle at the top left is \(67.4^\circ\), then the side \(x\) is adjacent to \(67.4^\circ\), and the hypotenuse is 26. Wait, but let's check with the initial wrong equation: \(x = \frac{26}{\cos(67.4^\circ)}\). Let's compute that. \(\cos(67.4^\circ) \approx 0.386287\), so \(26 / 0.386287 \approx 67.31\), which is too big. Wait, that can't be. Wait, maybe the hypotenuse is not 26? Wait, no, the triangle shows the hypotenuse as 26. Wait, maybe the angle is at the other acute angle? Wait, no, the angle is labeled at the top left as \(67.4^\circ\). Wait, perhaps the initial mistake was in the trigonometric ratio: it should be \(\sin(67.4^\circ)=\frac{x}{26}\)? Wait, no, \(\sin\) is opposite over hypotenuse. If \(x\) is the opposite side to \(67.4^\circ\), then \(\sin(67.4^\circ)=\frac{x}{26}\), so \(x = 26\sin(67.4^\circ)\). Let's calculate that. \(\sin(67.4^\circ) \approx 0.9223\), so \(26\times0.9223 \approx 23.98\). Wait, that's closer. Wait, maybe I misidentified the adjacent and opposite sides. Let's draw the triangle: right angle at bottom left (let's call that point \(C\)), top left is \(A\) (angle \(67.4^\circ\)), bottom right is \(B\). So sides: \(AC = x\) (vertical leg), \(BC\) (horizontal leg), \(AB = 26\) (hypotenuse). Angle at \(A\) is \(67.4^\circ\). So in triangle \(ABC\), right-angled at \(C\), angle at \(A\) is \(67.4^\circ\). Then:

- Adjacent to angle \(A\): \(AC = x\)
- Opposite to angle \(A\): \(BC\)
- Hypotenuse: \(AB = 26\)

Therefore, \(\cos(\angle A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{x}{26}\)

\(\sin(\angle A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB}\)

So the initial equation was wrong: it should be \(\cos(67.4^\circ) = \frac{x}{26}\), not \(\frac{26}{x}\). Therefore, the correct equation is \(x = 26\cos(67.4^\circ)\). Wait, but when we calculate that, \(\cos(67.4^\circ) \approx 0.386287\), so \(26 \times 0.386287 \approx 10.04\). But that seems small. Wait, maybe the angle is at \(B\)? Wait, no, the angle is labeled at \(A\) (top left) as \(67.4^\circ\). Wait, maybe the hypotenuse is \(x\)? No, the hypotenuse is the side opposite the right angle, so it can't be \(x\) because \(x\) is a leg. Wait, I think the initial mistake was in the trigonometric ratio: the person set up \(\cos(67.4^\circ) = \frac{\text{hypotenuse}}{\text{adjacent}}\), which is incorrect. Cosine is adjacent over hypotenuse, so the correct setup is \(\cos(67.4^\circ) = \frac{x}{26}\), so \(x = 26\cos(67.4^\circ)\). Let's compute that:

\(\cos(67.4^\circ) \approx \cos(67.4) \approx 0.386287\)

\(x = 26 \times 0.386287 \approx 10.04\)

Wait, but let's check with the initial wrong equation: \(x = \frac{26}{\cos(67.4^\circ)} \approx \frac{26}{0.386287} \approx 67.31\), which is impossible because the hypotenuse is 26, so a leg can't be longer than the hypotenuse. Ah! That's the key. In a right triangle, the legs are shorter than the hypotenuse. So \(x\) is a leg, so it must be shorter than 26. Therefore, the initial equation was wrong because it would make \(x\) longer than the hypotenuse (since \(\cos(67.4^\circ) < 1\), so \(26 / \cos(67.4^\circ) > 26\)), which is impossible. Therefore, the correct equation is \(\cos(67.4^\circ) = \frac{x}{26}\), so \(x = 26\cos(67.4^\circ) \approx 10.04\).

Wait, but let's confirm with the angle. If \(x = 10.04\), then \(\cos(67.4^\circ) = 10.04 / 26 \approx 0.386\), which matches the cosine value. So that's correct.

Step3: Calculate the correct value of \(x\)

Using the correct trigonometric ratio: \(\cos(67.4^\circ) = \frac{x}{26}\)

Solving for \(x\): \(x = 26 \times \cos(67.4^\circ)\)

Using a calculator to find \(\cos(67.4^\circ)\) (in degree mode):

\(\cos(67.4^\circ) \approx 0.386287\)

Then \(x = 26 \times 0.386287 \approx 10.04\) (rounded to two decimal places)

Wait, but let's check again. Wait, maybe the angle is \(67.4^\circ\), and the side \(x\) is the adjacent side, so \(\cos\) is correct, but the initial setup was inverted. So the error was in the trigonometric ratio setup (adjacent and hypotenuse were inverted). Therefore, the correct \(x\) is \(26 \times \cos(67.4^\circ) \approx 10.04\).

Wait, but I'm confused because when I think of the angle, if the hypotenuse is 26, and the leg is \(x\), then \(x\) must be less than 26, which 10.04 is. The initial wrong equation gave \(x \approx 67.31\), which is more than 26, impossible for a leg in a right triangle. So the correct equation is \(\cos(67.4^\circ) = \frac{x}{26}\), so \(x = 26 \cos(67.4^\circ) \approx 10.04\).

Answer:

\(10.04\)