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The problem asks us to determine the exact values of \(\cos(a)\) and \(\tan(a)\), given that \(\sin(a) = \frac{1}{2}\) and that \(a\) is in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).

### Step 1: Find \(\cos(a)\)
We can use the Pythagorean identity:

\[
\sin^2(a) + \cos^2(a) = 1
\]

Given that \(\sin(a) = \frac{1}{2}\), we substitute this into the identity:

\[
\left(\frac{1}{2}\right)^2 + \cos^2(a) = 1
\]

\[
\frac{1}{4} + \cos^2(a) = 1
\]

\[
\cos^2(a) = 1 - \frac{1}{4} = \frac{3}{4}
\]

\[
\cos(a) = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
\]

Since \(a\) is in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), which corresponds to the first and fourth quadrants, and in this interval, the cosine function is positive for angles in the first quadrant and negative for angles in the fourth quadrant. Since \(\sin(a)\) is positive (\(\frac{1}{2}\)), we know that \(a\) is in the first quadrant, so \(\cos(a)\) must be positive. Therefore,

\[
\cos(a) = \frac{\sqrt{3}}{2}
\]

### Step 2: Find \(\tan(a)\)
The tangent function is defined as:

\[
\tan(a) = \frac{\sin(a)}{\cos(a)}
\]

Substituting the known values of \(\sin(a) = \frac{1}{2}\) and \(\cos(a) = \frac{\sqrt{3}}{2}\):

\[
\tan(a) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

### Final Answer:
\[
\cos(a) = \frac{\sqrt{3}}{2}, \quad \tan(a) = \frac{\sqrt{3}}{3}
\]

Would you like a further explanation or have any other questions?

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Here are five related questions to explore this further:
1. How do the signs of \(\sin(a)\), \(\cos(a)\), and \(\tan(a)\) vary in different quadrants?
2. Can you derive \(\sec(a)\), \(\csc(a)\), and \(\cot(a)\) from these values?
3. What is the geometric interpretation of these values on the unit circle?
4. How does the Pythagorean identity relate to the unit circle?
5. What is the relationship between the sine and cosine functions for complementary angles?

**Tip:** Always check the quadrant when determining the sign of trigonometric functions to ensure the correct solution.

For a given angle a in the interval [-π/2, π/2], we have sin(a) = 1/2. Determine cos(a) and tan(a) and give their exact values.

Learn how to determine the exact values of cos(a) and tan(a) when sin(a) = 1/2 in the interval [-π/2, π/2]. This solution involves key trigonometric concepts, such as the Pythagorean Identity and tangent definition. Suitable for Grades 9-12.

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