Question

right triangle abc has its right angle at c. ac = 12 and bc = 5. which trigonometric ratios are correct? select each correct answer. sin b = 12/13 tan b = 5/12 tan a = 12/13 cos a = 12/13 sin a = 5/13

Explanation:

Step1: Find the length of the hypotenuse \(AB\) using the Pythagorean theorem.

By the Pythagorean theorem \(AB^{2}=AC^{2}+BC^{2}\). Given \(AC = 12\) and \(BC = 5\), then \(AB=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\).

Step2: Recall the definitions of trigonometric ratios.

The sine of an angle in a right - triangle is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), the cosine is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), and the tangent is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For \(\angle B\):

• \(\sin B=\frac{AC}{AB}=\frac{12}{13}\) (opposite side to \(\angle B\) is \(AC\) and hypotenuse is \(AB\)).
• \(\tan B=\frac{AC}{BC}=\frac{12}{5}\) (opposite side to \(\angle B\) is \(AC\) and adjacent side is \(BC\)).

For \(\angle A\):

• \(\tan A=\frac{BC}{AC}=\frac{5}{12}\) (opposite side to \(\angle A\) is \(BC\) and adjacent side is \(AC\)).
• \(\cos A=\frac{AC}{AB}=\frac{12}{13}\) (adjacent side to \(\angle A\) is \(AC\) and hypotenuse is \(AB\)).
• \(\sin A=\frac{BC}{AB}=\frac{5}{13}\) (opposite side to \(\angle A\) is \(BC\) and hypotenuse is \(AB\)).

Answer:

\(\sin B=\frac{12}{13}\), \(\cos A=\frac{12}{13}\), \(\sin A=\frac{5}{13}\)