Question

trigonometric ratios
________ ratios are functions that take an acute ____ measure as the input
and output a ratio of side lengths of a ______ triangle. three trigonometric ratios are sine,
cosine, and tangent.
\\( \sin(b)=\frac{\overline{\quad\quad}}{ab} \\)
\\( \quad\quad\cdot \frac{\text{opposite}}{\text{hypotenuse}} \\)
\\( \cos(b)=\frac{bc}{\overline{\quad\quad}} \\)
\\( \quad\quad\cdot \frac{\text{adjacent}}{\text{hypotenuse}} \\)
\\( \tan(b)=\frac{ac}{\overline{\quad\quad}} \\)
\\( \quad\quad\cdot \frac{\text{opposite}}{\text{adjacent side}} \\)

Explanation:

Step1: Analyze Trigonometric Ratios Definition

Trigonometric ratios are functions related to right - angled triangles. The first blank: The ratios are "Trigonometric" ratios. The second blank: They take an acute "angle" measure as input. The third blank: They are for a "right - angled" (or "right") triangle.

Step2: Analyze \(\sin(B)\)

In a right - triangle, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle \(B\), the opposite side to \(B\) is \(AC\), and the hypotenuse is \(AB\). So \(\sin(B)=\frac{AC}{AB}\).

Step3: Analyze \(\cos(B)\)

\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle \(B\), the adjacent side is \(BC\) and the hypotenuse is \(AB\). So \(\cos(B)=\frac{BC}{AB}\), and the blank in the denominator is \(AB\).

Step4: Analyze \(\tan(B)\)

\(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). For angle \(B\), the opposite side is \(AC\) and the adjacent side is \(BC\). So \(\tan(B)=\frac{AC}{BC}\), and the blank in the denominator is \(BC\).

Answer:

1. First blank: Trigonometric; Second blank: angle; Third blank: right (or right - angled)
2. \(\sin(B)=\frac{AC}{AB}\)
3. \(\cos(B)=\frac{BC}{AB}\) (denominator: \(AB\))
4. \(\tan(B)=\frac{AC}{BC}\) (denominator: \(BC\))