Question

1. lin missed class, and tyler is helping her use this table to approximate the angle measures that correspond to the ratios listed.

angle	adjacent leg ÷ hypotenuse	opposite leg ÷ hypotenuse	opposite leg ÷ adjacent leg
	0.819	0.573	0.700

tyler says, \you can use the right triangle table to figure this out.\ lin notices that some of the ratios are the same in each row. estimate the angles, and explain why some of the values are repeated.

Explanation:

Step1: Identify trigonometric ratios

Recall that for a right triangle:

• $\cos(\theta) = \frac{\text{Adjacent Leg}}{\text{Hypotenuse}}$
• $\sin(\theta) = \frac{\text{Opposite Leg}}{\text{Hypotenuse}}$
• $\tan(\theta) = \frac{\text{Opposite Leg}}{\text{Adjacent Leg}}$

Step2: Calculate first angle

Use inverse cosine for $\cos(\theta)=0.573$:
$\theta = \cos^{-1}(0.573) \approx 55^\circ$
Verify: $\sin(55^\circ) \approx 0.819$, $\tan(55^\circ) \approx 1.428$, which matches the first row.

Step3: Calculate second angle

Use inverse cosine for $\cos(\theta)=0.819$:
$\theta = \cos^{-1}(0.819) \approx 35^\circ$
Verify: $\sin(35^\circ) \approx 0.573$, $\tan(35^\circ) \approx 0.700$, which matches the second row.

Step4: Explain repeated ratios

In a right triangle, the two acute angles are complementary: $\theta + (90^\circ - \theta) = 90^\circ$. By co-function identities:
$\sin(\theta) = \cos(90^\circ - \theta)$, $\cos(\theta) = \sin(90^\circ - \theta)$
So the sine of one angle equals the cosine of its complementary angle.

Answer:

1. The angles are approximately $55^\circ$ (for the first row of ratios) and $35^\circ$ (for the second row of ratios).
2. The ratios repeat because the two acute angles in a right triangle are complementary (they add to $90^\circ$). The sine of one angle is equal to the cosine of its complementary angle, which is why the adjacent/hypotenuse ratio of one angle matches the opposite/hypotenuse ratio of the other angle.