Question

1. in the triangle below, compare and contrast the trig ratios for angle x and angle z. can you make any connections between them?

Explanation:

Identify the side lengths relative to angles X and Z

$$
LATEXBLOCK0
$$

Calculate the trigonometric ratios for both angles

$$
LATEXBLOCK1
$$

Compare and identify connections

$$
LATEXBLOCK2
$$

Answer:

The trigonometric ratios for angle \(X\) and angle \(Z\) are:

• Angle \(X\):

$$ \sin(X) = \frac{\sqrt{11}}{6}, \quad \cos(X) = \frac{5}{6}, \quad \tan(X) = \frac{\sqrt{11}}{5} $$

• Angle \(Z\):

$$ \sin(Z) = \frac{5}{6}, \quad \cos(Z) = \frac{\sqrt{11}}{6}, \quad \tan(Z) = \frac{5\sqrt{11}}{11} $$

Connections:

1. Cofunction Relationships: The sine of one acute angle is equal to the cosine of the other acute angle because they are complementary angles (\(X + Z = 90^\circ\)):

$$ \sin(X) = \cos(Z) = \frac{\sqrt{11}}{6} \quad \text{and} \quad \cos(X) = \sin(Z) = \frac{5}{6} $$

1. Reciprocal Tangents: The tangent values are reciprocals of each other:

$$ \tan(X) \cdot \tan(Z) = 1 $$