The problem asks to find which trigonometric ratio is equivalent to a trigonometric ratio of an angle less than 45 degrees. The options are sin(63°), cos(82°), and tan(73°).

1. Problem Description

The problem asks to find which trigonometric ratio is equivalent to a trigonometric ratio of an angle less than 45 degrees. The options are sin(63°), cos(82°), and tan(73°).

2. Solution Steps

We need to use the following trigonometric identities:

$sin(\theta) = cos(90^{\circ}-\theta)$

$cos(\theta) = sin(90^{\circ}-\theta)$

$tan(\theta) = \frac{1}{tan(90^{\circ}-\theta)}$ if $\theta < 45^{\circ}$

Let's analyze each option:

a. sin(63°): Using the identity $sin(\theta) = cos(90^{\circ}-\theta)$, we have $sin(63^{\circ}) = cos(90^{\circ} - 63^{\circ}) = cos(27^{\circ})$. Since $27^{\circ} < 45^{\circ}$, this is a possible solution.

b. cos(82°): Using the identity $cos(\theta) = sin(90^{\circ}-\theta)$, we have $cos(82^{\circ}) = sin(90^{\circ} - 82^{\circ}) = sin(8^{\circ})$. Since $8^{\circ} < 45^{\circ}$, this is a possible solution.

c. tan(73°): We can write $tan(73^{\circ})$ as $\frac{1}{tan(90^{\circ} - 73^{\circ})} = \frac{1}{tan(17^{\circ})}$. Since $17^{\circ} < 45^{\circ}$, this is also a possible solution. However, it does not return a basic trigonometric function. In other words, this is not equivalent to $sin(x)$, $cos(x)$ or $tan(x)$ for some $x < 45$.

Since we need to find WHICH trigonometric ratio is equivalent to a trigonometric ratio of an angle less than 45 degrees, we need to look at sin(63°) and cos(82°). Both have angles less than 45 after conversion. However, tan(73°) does not lead to something equivalent to the tan of an angle less than 45 without reciprocals. If the problem is asking which of the given ratios is equal to either sin, cos or tan of an angle less than 45 degrees, the answers are:

sin(63°) = cos(27°)

cos(82°) = sin(8°)

tan(73°) = 1/tan(17°)

Both a and b are valid options. Let us analyze further.

$sin(63^{\circ})$ = $cos(27^{\circ})$

$cos(82^{\circ})$ = $sin(8^{\circ})$

The question probably asks to find THE simplest trigonometric ratio that is less than $45^{\circ}$, which would be $sin(8^{\circ})$.

3. Final Answer

cos(82°)