QuestionFind the value of $\cot \frac{3 \pi}{2} + \cos \frac{3 \pi}{2}$ without a calculator.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the exact value of the expression $\cot \frac{3 \pi}{}+\cos \frac{3 \pi}{}$ . We are not allowed to use a calculator.
3. We need to use the definitions of cotangent and cosine in terms of the unit circle.

STEP 2

First, let's find the value of $\cot \frac{ \pi}{2}$ The cotangent function is the reciprocal of the tangent function. In terms of the unit circle, the tangent of an angle is the y-coordinate divided by the x-coordinate. Therefore, the cotangent of an angle is the x-coordinate divided by the y-coordinate.

STEP 3

For an angle of $\frac{3 \pi}{2}$ radians, the point on the unit circle is (0, -1). Therefore, the x-coordinate is0 and the y-coordinate is -1.

STEP 4

Substitute these values into the definition of cotangent to find $\cot \frac{3 \pi}{2}$ $\cot \frac{3 \pi}{2} = \frac{x}{y} = \frac{0}{-1}$

STEP 5

Calculate the value of $\cot \frac{3 \pi}{2}$ $\cot \frac{3 \pi}{2} =0$

STEP 6

Next, let's find the value of $\cos \frac{3 \pi}{2}$ The cosine of an angle is the x-coordinate of the point on the unit circle.

STEP 7

For an angle of $\frac{3 \pi}{2}$ radians, the point on the unit circle is (0, -1). Therefore, the x-coordinate is0.

STEP 8

Substitute this value into the definition of cosine to find $\cos \frac{3 \pi}{2}$ $\cos \frac{3 \pi}{2} = x =0$

STEP 9

Now that we have the values of $\cot \frac{3 \pi}{2}$ and $\cos \frac{3 \pi}{2}$ , we can substitute these into the original expression.
$\cot \frac{3 \pi}{2}+\cos \frac{3 \pi}{2} = +$

STEP 10

Calculate the value of the original expression.
$\cot \frac{3 \pi}{2}+\cos \frac{3 \pi}{2} =0$ The exact value of the expression $\cot \frac{3 \pi}{2}+\cos \frac{3 \pi}{2}$ is0.

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