Question

1. noah missed class, and clare is helping him use the table to approximate the angle measures that have the ratios listed. clare says, \you can use the right triangle table to figure this out.\ noah notices that some of the ratios are the same in each row. estimate the angles and explain why some of the values are repeated.

angle	adjacent leg ÷ hypotenuse	opposite leg ÷ hypotenuse	opposite leg ÷ adjacent leg
	0.174	0.985	5.671

Explanation:

Step1: Match ratio to known angle

We recognize $\cos(\theta)=0.985$, $\sin(\theta)=0.174$ corresponds to $\theta \approx 10^\circ$, since $\cos(10^\circ)\approx0.9848$, $\sin(10^\circ)\approx0.1736$.

Step2: Identify complementary angle

For the second row, $\cos(\alpha)=0.174$, $\sin(\alpha)=0.985$. This is $\alpha \approx 80^\circ$, as $\cos(80^\circ)\approx0.1736$, $\sin(80^\circ)\approx0.9848$.

Step3: Explain repeated ratio logic

$\sin(\theta)=\cos(90^\circ-\theta)$ and $\cos(\theta)=\sin(90^\circ-\theta)$. $10^\circ$ and $80^\circ$ are complementary ($10^\circ+80^\circ=90^\circ$), so their sine/cosine swap.

Step4: Verify tangent values

For $10^\circ$: $\tan(10^\circ)=\frac{\sin(10^\circ)}{\cos(10^\circ)}\approx\frac{0.1736}{0.9848}\approx0.176$. For $80^\circ$: $\tan(80^\circ)=\frac{\sin(80^\circ)}{\cos(80^\circ)}\approx\frac{0.9848}{0.1736}\approx5.671$.

Answer:

• The first row corresponds to an angle of approximately $\boldsymbol{10^\circ}$.
• The second row corresponds to an angle of approximately $\boldsymbol{80^\circ}$.
• The repeated (swapped) ratios occur because $10^\circ$ and $80^\circ$ are complementary angles. For any acute angle $\theta$, $\sin(\theta)=\cos(90^\circ-\theta)$ and $\cos(\theta)=\sin(90^\circ-\theta)$: the sine of one angle equals the cosine of its complement, which swaps the adjacent/hypotenuse and opposite/hypotenuse ratios between the two angles.