Question

1. match the following.

a) opposite leg to \\( \angle a \\)
b) sine ratio of \\( \angle c \\)
c) opposite angle to \\( \overline{ab} \\)
d) the hypotenuse
e) adjacent leg to \\( \angle a \\)
f) tangent ratio of \\( \angle c \\)
g) reference angle if \\( \frac{bc}{ac} \\) is the cosine ratio.
h) adjacent leg to \\( \angle c \\)
i) cosine ratio of \\( \angle a \\)
j) reference angle if \\( \frac{bc}{ac} \\) is the sine ratio.

Explanation:

Step1: Recall trigonometric ratios and triangle side - angle relationships

• Sine ratio: $\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine ratio: $\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent ratio: $\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}$
• In a right - triangle, the hypotenuse is the longest side.

Step2: Match each part

• a) Opposite leg to $\angle A$: $\overline{BC}$ (because in right - triangle $\triangle ABD$ (assuming the right - angle is at $D$), for angle $A$, the side opposite is $BC$)
• b) Sine ratio of $\angle C$: $\frac{AB}{AC}$ (since $\sin C=\frac{\text{Opposite to }C}{\text{Hypotenuse}}$, and opposite to $C$ is $AB$ and hypotenuse is $AC$)
• c) Opposite Angle to $\overline{AB}$: $\angle C$ (by the property that in a triangle, the side is opposite to the angle)
• d) The Hypotenuse: $\overline{AC}$ (in a right - triangle, the hypotenuse is the side opposite the right - angle. If the right - angle is at $B$, then $AC$ is the hypotenuse)
• e) Adjacent leg to $\angle A$: $\overline{AB}$ (using the definition of adjacent side for an angle in a right - triangle)
• f) Tangent ratio of $\angle C$: $\frac{AB}{BC}$ (since $\tan C=\frac{\text{Opposite to }C}{\text{Adjacent to }C}$, opposite to $C$ is $AB$ and adjacent to $C$ is $BC$)
• g) Reference angle if $\frac{BC}{AC}$ is the Cosine Ratio: $\angle C$ (because $\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}$, if $\frac{BC}{AC}$ is the cosine ratio, then adjacent side is $BC$ and hypotenuse is $AC$, so the angle is $C$)
• h) Adjacent leg to $\angle C$: $\overline{BC}$ (by the definition of adjacent side for an angle in a right - triangle)
• i) Cosine Ratio of $\angle A$: $\frac{AB}{AC}$ (since $\cos A=\frac{\text{Adjacent to }A}{\text{Hypotenuse}}$, adjacent to $A$ is $AB$ and hypotenuse is $AC$)
• j) Reference angle if $\frac{BC}{AC}$ is the Sine Ratio: $\angle A$ (because $\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}$, if $\frac{BC}{AC}$ is the sine ratio, then opposite side is $BC$ and hypotenuse is $AC$, so the angle is $A$)

Answer:

a) $\overline{BC}$
b) $\frac{AB}{AC}$
c) $\angle C$
d) $\overline{AC}$
e) $\overline{AB}$
f) $\frac{AB}{BC}$
g) $\angle C$
h) $\overline{BC}$
i) $\frac{AB}{AC}$
j) $\angle A$