Question

calculating trigonometric ratios
determine the value of sin(b) for δabc.

• sin(30)=\frac{opposite}{hypotenuse}

=\frac{10}{\\_\\_\\_}=\frac{\\_\\_\\_}{2}

defining trigonometric ratios
the three basic trigonometric functions for the acute angles of a right triangle are sine,
\\_\\_\\_\\_\\_\\_, and tangent. the functions relate a given angle measure and a ratio of
side lengths.

• cos(a)=\frac{adjacent}{hypotenuse}=\frac{ac}{\\_\\_\\_}
• cosine of angle a = cos(a)

Explanation:

Step1: Recall SOHCAHTOA

The three basic trigonometric functions for acute angles in a right triangle are sine, cosine, and tangent (from SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent). So the missing function is cosine.

Step2: Calculate sin(B)

For \(\triangle ABC\), angle \(B = 30^\circ\), right - angled at \(C\). The formula for sine is \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle \(B\), the side opposite to \(B\) is \(AC = 10\), and the hypotenuse \(AB=20\). So \(\sin(30^{\circ})=\frac{10}{20}=\frac{1}{2}\). Also, for the \(\cos(A)\) formula, \(AC = 10\) and \(AB = 20\), so \(\cos(A)=\frac{AC}{AB}=\frac{10}{20}=\frac{1}{2}\) (but we focus on \(\sin(B)\) first).

Step3: Fill in the blanks

• For the trigonometric functions blank: The three basic functions are sine, cosine, and tangent. So the blank is filled with "cosine".
• For \(\sin(30^{\circ})=\frac{10}{\_\_}\), since \(\sin(30^{\circ})=\frac{10}{20}\), the blank is \(20\).
• For \(\cos(A)=\frac{AC}{\_\_}\), since \(AC = 10\) and \(AB = 20\) (hypotenuse), \(\cos(A)=\frac{AC}{AB}=\frac{10}{20}\), so the blank is \(AB\) (or \(20\)).

Answer:

• The missing trigonometric function: cosine
• For \(\sin(30^{\circ})=\frac{10}{\_\_}\), the answer is \(20\)
• For \(\cos(A)=\frac{AC}{\_\_}\), the answer is \(AB\) (or \(20\))
• \(\sin(B)=\frac{1}{2}\)