Question

trigonometric ratios
which trigonometric ratios are correct for triangle abc? choose three correct answers.
sin(c) = \frac{\sqrt{3}}{2}
tan(b) = \frac{2\sqrt{3}}{3}
tan(c) = \sqrt{3}
sin(b) = \frac{1}{2}
cos(b) = \frac{\sqrt{3}}{2}
(image of right triangle abc with right angle at a, angle at b is 30°, angle at c is 60°, side ac = 9, side bc = 18)

Explanation:

First, let's recall the trigonometric ratios: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In triangle \(ABC\), \(\angle A = 90^\circ\), \(\angle B = 30^\circ\), \(\angle C = 60^\circ\), \(AC = 9\), \(AB\) (let's find it: in a 30-60-90 triangle, sides are in ratio \(1:\sqrt{3}:2\). Hypotenuse \(BC = 18\), so the side opposite \(30^\circ\) (which is \(AC\)) is \(9\), so \(AB = 9\sqrt{3}\) (opposite \(60^\circ\)).

Step 1: Check \(\sin(C)\)

\(\angle C = 60^\circ\), \(\sin(60^\circ)=\frac{\sqrt{3}}{2}\). Opposite to \(C\) is \(AB = 9\sqrt{3}\), hypotenuse \(BC = 18\). So \(\sin(C)=\frac{AB}{BC}=\frac{9\sqrt{3}}{18}=\frac{\sqrt{3}}{2}\). So this is correct.

Step 2: Check \(\sin(B)\)

\(\angle B = 30^\circ\), \(\sin(30^\circ)=\frac{1}{2}\). Opposite to \(B\) is \(AC = 9\), hypotenuse \(BC = 18\). So \(\sin(B)=\frac{AC}{BC}=\frac{9}{18}=\frac{1}{2}\). Correct.

Step 3: Check \(\tan(C)\)

\(\angle C = 60^\circ\), \(\tan(60^\circ)=\sqrt{3}\). Opposite to \(C\) is \(AB = 9\sqrt{3}\), adjacent to \(C\) is \(AC = 9\). So \(\tan(C)=\frac{AB}{AC}=\frac{9\sqrt{3}}{9}=\sqrt{3}\). Correct.

Step 4: Check \(\tan(B)\)

\(\angle B = 30^\circ\), \(\tan(30^\circ)=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\). But the option says \(\tan(B)=\frac{2\sqrt{3}}{3}\), which is wrong.

Step 5: Check \(\cos(B)\) (if needed, but let's see the options). Wait, the options given: \(\sin(C)=\frac{\sqrt{3}}{2}\) (correct), \(\sin(B)=\frac{1}{2}\) (correct), \(\tan(C)=\sqrt{3}\) (correct). The other two: \(\tan(B)\) is wrong, and let's check \(\cos(B)\) (if there was an option, but from the visible options, the three correct are \(\sin(C)=\frac{\sqrt{3}}{2}\), \(\sin(B)=\frac{1}{2}\), \(\tan(C)=\sqrt{3}\).

Answer:

The correct trigonometric ratios are:

• \(\sin(C)=\frac{\sqrt{3}}{2}\)
• \(\sin(B)=\frac{1}{2}\)
• \(\tan(C)=\sqrt{3}\)