Given that angle C is a right angle in triangle ABC, the statement from the given options that will always be true considering their trigonometry ratios is:
B. Sin(A) = Cos(B)
Recall:
- Trigonometry ratios are: SOH CAH TOA.
- They can be used when solving a right triangle.
The image of the right triangle ABC is shown in the attachment below.
We are given the following:
AC = 5
AB = 13
BC = 12
Let's find the trigonometry ratios for angle A and angle B respectively.
Trigonometry ratios of angle A as the reference angle:
[tex]Sin(A) = \frac{Opp}{Hyp} = \frac{12}{13}[/tex]
[tex]Cos(A) = \frac{Adj}{Hyp} = \frac{5}{13}[/tex]
[tex]Tan(A) = \frac{Opp}{Adj} = \frac{12}{5}[/tex]
Trigonometry ratios of angle B as the reference angle:
[tex]Sin(B) = \frac{Opp}{Hyp} = \frac{5}{13}[/tex]
[tex]Cos(B) = \frac{Adj}{Hyp} = \frac{12}{13}[/tex]
[tex]Tan(B) = \frac{Opp}{Adj} = \frac{5}{12}[/tex]
From the above, we can deduce the following:
- Sin(A) = Cos(B)
- Cos(A) = Sin(B)
Therefore, given that angle C is a right angle in triangle ABC, the statement from the given options that will always be true considering their trigonometry ratios is:
B. Sin(A) = Cos(B)
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