Question

defining trigonometric ratios
the three basic trigonometric functions for the ______ angles of a right triangle are sine,
cosine, and tangent. each function relates a given angle measure to a particular ratio of
side lengths.

• these ratios will only be true for ______ triangles.
• a function is relation in which each element of the domain is mapped to or paired with

exactly one element of the range.

• input -- ______ measure
• output -- ______ of side lengths
• the three ratios are true for specific angles of any right triangle, because those

triangles with the same angles will be ______ triangles.
label the part of the function that is the \input\ and \output\.

• $sin(a)=\frac{\text{opposite}}{\text{hypotenuse}}$

$=\frac{\text{bc}}{\\_\\_\\_}$

Explanation:

Step1: Analyze the first blank

Trigonometric ratios (sine, cosine, tangent) are defined for the acute angles of a right triangle (since the right angle is \(90^\circ\), and the other two are acute, i.e., less than \(90^\circ\)).

Step2: Analyze the second blank

These ratios (trigonometric) hold true only for right triangles (by definition of trigonometric ratios in right - angled triangles).

Step3: Analyze the third blank (Input)

In a trigonometric function (like \(\sin(A)\), \(\cos(A)\), \(\tan(A)\)), the input is the angle measure (e.g., the measure of angle \(A\) in the right triangle).

Step4: Analyze the fourth blank (Output)

The output of a trigonometric function is the ratio of the lengths of two sides of the right triangle (e.g., \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}\) is a ratio of side lengths).

Step5: Analyze the fifth blank

Triangles with the same angles are similar triangles (by the AA - Angle - Angle similarity criterion; in right triangles, if two angles are equal, the third is also equal, so they are similar).

Step6: Analyze the labels for \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{\_\_\_}\)

In the right triangle \(ABC\) (right - angled at \(C\)), the hypotenuse is \(AB\). So \(\sin(A)=\frac{BC}{AB}\). Also, for the function \(\sin(A)\), the input is the angle \(A\) (so the first label (left speech bubble) is "Angle \(A\) (angle measure)" and the output is the ratio \(\frac{\text{opposite}}{\text{hypotenuse}}\) (or the specific ratio \(\frac{BC}{AB}\)).

Answer:

1. First blank: acute
2. Second blank: right
3. Third blank: angle
4. Fourth blank: ratio
5. Fifth blank: similar
6. For \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{\_\_\_}\): \(AB\)
7. Labels for \(\sin(A)\): Left speech bubble: Angle \(A\) (angle measure), Right speech bubble: Ratio of side lengths (or \(\frac{\text{opposite}}{\text{hypotenuse}}\))