Question

Find the value of the trigonometric function. If possible, give the exact value; otherwise, use a calculator to find an approximate value correct to five decimal places. tan 5π/6

Answer 1

The exact value of tan 5π/6 is found using reference angle properties, yielding -√3/3 after simplifying. This value is negative, as expected for tangent values in the second quadrant.

Step-by-step explanation:

The value of the trigonometric function tan 5π/6 can be found by considering the properties of the unit circle and the reference angles. Since 5π/6 is in the second quadrant where tangent values are negative and it is equivalent to π - π/6, we can use the reference angle π/6 to find the tangent value.

Using the idea that tan(θ) = sin(θ)/cos(θ), we have:

• sin(5π/6) = sin(π - π/6) = sin(π/6) = 1/2
• cos(5π/6) = cos(π - π/6) = -cos(π/6) = -√3/2

So:

tan(5π/6) = sin(5π/6) / cos(5π/6) = (1/2) / (-√3/2) = -1/√3

After simplifying, we get the exact value:

tan(5π/6) = -1/√3 = -√3/3

To check if our answer is reasonable, we examine the properties of the unit circle and the function of tangent in the specified quadrant, which confirms that the obtained value is indeed reasonable, as the tangent of an angle in the second quadrant should be negative.