To solve for the sine of the other acute angle in a right triangle where the tangent of one of the acute angles is [tex]\(\frac{\sqrt{5}}{4}\)[/tex], follow these steps:
### Step 1: Define the Given Tangent
In the right triangle, let’s denote the given acute angle as [tex]\( \theta \)[/tex]. We know:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{5}}{4} \][/tex]
### Step 2: Identify Triangle Sides
From the tangent definition, assign the lengths of the triangle sides. Suppose:
- The length of the opposite side is [tex]\( \sqrt{5} \)[/tex]
- The length of the adjacent side is [tex]\( 4 \)[/tex]
### Step 3: Find the Length of the Hypotenuse
Using the Pythagorean theorem to find the hypotenuse ([tex]\(h\)[/tex]):
[tex]\[ h = \sqrt{(\sqrt{5})^2 + 4^2} = \sqrt{5 + 16} = \sqrt{21} \][/tex]
### Step 4: Calculate Sine of the Angle [tex]\( \theta \)[/tex]
By definition, the sine of [tex]\( \theta \)[/tex] is:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{5}}{\sqrt{21}} \][/tex]
### Step 5: Use the Complementary Angle
In a right triangle, the two acute angles are complementary. Thus, if [tex]\( \theta \)[/tex] is one acute angle, [tex]\( \phi \)[/tex] is the other acute angle such that:
[tex]\[ \theta + \phi = 90^\circ \][/tex]
[tex]\[ \sin(\phi) = \cos(\theta) \][/tex]
Since:
[tex]\[ \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \][/tex]
We use:
[tex]\[ \cos(\theta) = \sqrt{1 - \left(\frac{\sqrt{5}}{\sqrt{21}}\right)^2} = \sqrt{1 - \frac{5}{21}} = \sqrt{\frac{21 - 5}{21}} = \sqrt{\frac{16}{21}} = \frac{4}{\sqrt{21}} \][/tex]
### Step 6: Calculate Sine of the Other Acute Angle
The sine of angle [tex]\(\phi\)[/tex] is the same as [tex]\( \cos(\theta) \)[/tex]:
[tex]\[ \sin(\phi) = \cos(\theta) = \frac{4}{\sqrt{21}} \][/tex]
### Step 7: Simplify and Match with Given Choices
To ensure it matches the multiple-choice format:
[tex]\[ \sin(\phi) = \frac{4 \cdot \sqrt{21}}{21} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{4 \sqrt{21}}{21}} \][/tex]
Which corresponds to option (A).
