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Select the true statement about triangle ABC.

Check the final answer first, then review the worked steps.

Problem

Select the true statement about triangle ABC.

Answer

\(A. \cos A = \sin C\)

Step-by-step solution

  1. Identify the given information: The problem provides a right-angled triangle ABC, with the right angle at B. The lengths of the sides are given as BC = 5, AB = 12, and AC = 13. We need to determine which of the given trigonometric statements is true.

2. Recall trigonometric ratios in a right-angled triangle: For an acute angle in a right-angled triangle:
- Sine (sin) = Opposite / Hypotenuse
- Cosine (cos) = Adjacent / Hypotenuse
- Tangent (tan) = Opposite / Adjacent

3. Calculate the trigonometric values for angle A:
- The side opposite to angle A is BC, which has a length of 5.
- The side adjacent to angle A is AB, which has a length of 12.
- The hypotenuse is AC, which has a length of 13.
Therefore:
- $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{12}{13}$
- $\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{5}{13}$
- $\tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB} = \frac{5}{12}$

4. Calculate the trigonometric values for angle C:
- The side opposite to angle C is AB, which has a length of 12.
- The side adjacent to angle C is BC, which has a length of 5.
- The hypotenuse is AC, which has a length of 13.
Therefore:
- $\cos C = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{5}{13}$
- $\sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{12}{13}$
- $\tan C = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{12}{5}$

5. Evaluate the given statements:
- A. $\cos A = \sin C$: We calculated $\cos A = \frac{12}{13}$ and $\sin C = \frac{12}{13}$. So, $\frac{12}{13} = \frac{12}{13}$. This statement is true.
- B. $\cos A = \cos C$: We calculated $\cos A = \frac{12}{13}$ and $\cos C = \frac{5}{13}$. So, $\frac{12}{13} \neq \frac{5}{13}$. This statement is false.
- C. $\cos A = \sin B$: Angle B is the right angle ($90^\circ$). The trigonometric functions for $90^\circ$ are $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$. $\cos A = \frac{12}{13}$, which is not equal to 1. This statement is false.
- D. $\cos A = \tan C$: We calculated $\cos A = \frac{12}{13}$ and $\tan C = \frac{12}{5}$. So, $\frac{12}{13} \neq \frac{12}{5}$. This statement is false.

  1. Conclusion: The only true statement is A.