Another: theta is a second quadrant angle in standard position and tan theta = - (sqrt5)/(2) .

Find the exact value of cos theta.

To find the exact value of cos(theta), we can use the given information that tan(theta) = -sqrt(5)/2.

Since theta is a second quadrant angle, we know that the tangent of theta is negative in the second quadrant.

We can use the relationship between the sine and cosine functions to find the cosine of theta.

Recall the identity: tan(theta) = sin(theta) / cos(theta).

Dividing both sides of the equation by sin(theta), we have:
tan(theta) / sin(theta) = sin(theta) / cos(theta) / sin(theta).

Using the fact that tan(theta) = -sqrt(5)/2, we can substitute this value into the left side of the equation:

(-sqrt(5)/2) / sin(theta) = sin(theta) / cos(theta) / sin(theta).

Next, we simplify the left side:

(-sqrt(5)/2) / sin(theta) = 1 / cos(theta)

Now we can cross multiply:

(-sqrt(5)/2) * cos(theta) = sin(theta)

Multiply both sides by -2 to eliminate the fraction:

sqrt(5) * cos(theta) = -2 * sin(theta)

Simplify the right side:

sqrt(5) * cos(theta) = -2sin(theta)

Recall the Pythagorean trigonometric identity:
sin^2(theta) + cos^2(theta) = 1

We can substitute the identities:
(sin^2(theta) + cos^2(theta)) / cos(theta) = 1 / cos(theta)

Using the given information that tan(theta) = -sqrt(5)/2, we can substitute this value into the left side of the equation:

(sin^2(theta) + cos^2(theta)) / cos(theta) = 1 / cos(theta)

Since we have an expression for sin(theta) in terms of cos(theta) from the previous step, we can substitute it into the equation:

(sin^2(theta) + cos^2(theta)) / cos(theta) = 1 / cos(theta)

(sin^2(theta) + (-2sin(theta))/sqrt(5))^2 + cos^2(theta)) / cos(theta) = 1 / cos(theta)

We can simplify this equation by combining like terms:

(sin^2(theta) + (4sin(theta))/sqrt(5) + (4/5) + cos^2(theta)) / cos(theta) = 1 / cos(theta)

(sin^2(theta) + (4sin(theta))/sqrt(5) + (4/5) + (1 - sin^2(theta))) / cos(theta) = 1 / cos(theta)

(1 + (4sin(theta))/sqrt(5) + (4/5) + (1 - 1)) / cos(theta) = 1 / cos(theta)

((4sin(theta))/sqrt(5) + (4/5) + 1) / cos(theta) = 1 / cos(theta)

Simplify the left side expression further:

(4sin(theta) + (4/5)sqrt(5) + sqrt(5)) / (sqrt(5) * cos(theta)) = 1 / cos(theta)

Now, since we know that tan(theta) = -sqrt(5)/2, we can substitute this value into the left side of the equation:

(4sin(theta) + (4/5)sqrt(5) + sqrt(5)) / (sqrt(5) * cos(theta)) = 1 / cos(theta)

Noticing the denominator on the left side, we can multiply the entire equation by sqrt(5) to eliminate the denominator:

4sin(theta) + (4/5)sqrt(5) + sqrt(5) = 1

Now, we can isolate the sin(theta) term:

4sin(theta) = 1 - (4/5)sqrt(5) - sqrt(5)

Simplify:

4sin(theta) = (5 - 4sqrt(5) - 5)/5

4sin(theta) = -4sqrt(5)/5

Divide both sides by 4:

sin(theta) = -sqrt(5)/5

Now that we have an expression for sin(theta), we can use the Pythagorean trigonometric identity to find cos(theta):

sin^2(theta) + cos^2(theta) = 1

Substituting the value of sin(theta) obtained above:

(-sqrt(5)/5)^2 + cos^2(theta) = 1

Simplify:

5/25 + cos^2(theta) = 1

1/5 + cos^2(theta) = 1

Subtract 1/5 from both sides:

cos^2(theta) = 1 - 1/5

cos^2(theta) = 4/5

To find the exact value of cos(theta), we take the square root of both sides:

cos(theta) = ± sqrt(4/5)

Since theta is a second quadrant angle, the cosine is negative in the second quadrant.

Therefore, the exact value of cos(theta) is:

cos(theta) = - sqrt(4/5)

Hence, the exact value of cos(theta) is - sqrt(4/5).