Question

find the first mistake in solving:
a right triangle has hypotenuse 13 and adjacent side 12.
find the missing angles.

1. use \\( \cos ( \theta ) = \frac { 12 } { 13 } \\).
2. compute \\( \cos ^ { - 1 } ( \frac { 12 } { 13 } ) = 22.62 ^ { \circ } \\).
3. conclude \\( \theta = 22.62 \\).
4. apply triangle sum theorem: \\( \theta + \alpha + 90 ^ { \circ } = 180 ^ { \circ } \\)
5. substitute value of \\( \theta : 22.62 + \alpha + 90 = 180 \\)
6. solve for \\( \alpha : \alpha + 135.24 = 180 \\)
7. conclude \\( \alpha = 112.62 ^ { \circ } \\).

Explanation:

Step1: Analyze line 1

Using \(\cos(\theta)=\frac{12}{13}\) is a valid use of the cosine ratio (\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)) in a right - triangle.

Step2: Analyze line 2

Computing \(\cos^{-1}(\frac{12}{13})\approx22.62^{\circ}\) is a correct calculation using the inverse cosine function.

Step3: Analyze line 3

Concluding \(\theta = 22.62\) is incorrect. The value of \(\theta\) should be in degrees, so it should be \(\theta=22.62^{\circ}\).

Step4: Analyze line 4

Applying the Triangle Sum Theorem \(\theta+\alpha + 90^{\circ}=180^{\circ}\) is correct for the sum of angles in a triangle.

Step5: Analyze line 5

Substituting \(\theta = 22.62^{\circ}\) (assuming the degree symbol was just omitted in line 3) into \(22.62+\alpha + 90 = 180\) (if we consider \(22.62\) as \(22.62^{\circ}\)) is a valid substitution.

Step6: Analyze line 6

Solving \(22.62+\alpha+90 = 180\) gives \(\alpha+112.62 = 180\) (not \(\alpha + 135.24=180\)). The error in line 6 is due to the incorrect value from line 3 (if we consider the non - degree issue in line 3). But the first mistake is in line 3.

Step7: Analyze line 7

Since the value of \(\theta\) was misrepresented in line 3 (no degree symbol), the subsequent calculation for \(\alpha\) is wrong. But the first mistake is in line 3.

Answer:

Line 3