Video Transcript
Find the exact value of the tan of seven 𝜋 over six without using a calculator.
We will begin by sketching the unit circle in order to identify the quadrant in which the angle seven 𝜋 over six lies in. We recall that any angle in standard position is measured from the positive 𝑥-axis. And if the angle is positive, as in this case, we measure in a counterclockwise direction. We can mark on the angles 𝜋 over two, 𝜋, three 𝜋 over two, and two 𝜋 radians. Seven 𝜋 over six is greater than 𝜋 but less than three 𝜋 over two. This means that the angle lies in the third quadrant.
We know that any point that lies on the unit circle has coordinates cos 𝜃, sin 𝜃. This means that the point at which the terminal side of our angle meets the unit circle has coordinates cos seven 𝜋 over six, sin seven 𝜋 over six. To work out the value of the 𝑥- and 𝑦-coordinates, we firstly need to find the measure of the angle between the terminal side and the 𝑥-axis, known as the reference angle. Since the angle between the positive 𝑥-axis and negative 𝑥-axis is 𝜋, the reference angle 𝛼 is equal to seven 𝜋 over six minus 𝜋, which is equal to 𝜋 over six.
By drawing a line perpendicular to the 𝑥-axis from the point of intersection, we can create a right triangle, as shown. We will now clear some space so we can draw an enlargement of this triangle. As the point lies on the unit circle, the length of the hypotenuse of our triangle is one unit. As already mentioned, the point lies in the third quadrant, which means that both the 𝑥- and 𝑦-coordinates of the point will be negative. Since the lengths of our triangles must be positive, we can multiply our coordinates by negative one, giving us side lengths of negative sin of seven 𝜋 over six and negative cos of seven 𝜋 over six.
In any right triangle, the sin of angle 𝜃 is equal to the opposite over the hypotenuse. And the cos of angle 𝜃 is equal to the adjacent over the hypotenuse. This means that the sin of 𝜋 over six is equal to negative sin of seven 𝜋 over six over one. 𝜋 over six radians or 30 degrees is one of our special angles. And we know that the sine of this angle is equal to one-half. This means that negative sin of seven 𝜋 over six is equal to one-half. And multiplying through by negative one, the sin of seven 𝜋 over six is equal to negative one-half.
We can repeat this using the cosine ratio. The cos of 𝜋 over six is equal to negative cos of seven 𝜋 over six divided by one, which is equal to negative cos of seven 𝜋 over six. The cos of 𝜋 over six or 30 degrees is equal to root three over two. So this is equal to negative cos seven 𝜋 over six. Once again, we can multiply through by negative one, giving us the cos of seven 𝜋 over six is equal to negative root three over two.
We have been asked to find the exact value of the tan of seven 𝜋 over six. And one of our trigonometric identities states that the tan of 𝜃 is equal to sin 𝜃 over cos 𝜃. The tan of seven 𝜋 over six is therefore equal to the sin of seven 𝜋 over six divided by the cos of seven 𝜋 over six. Using the values we have already calculated, this is equal to negative one-half divided by negative root three over two. We know that dividing by a fraction is the same as multiplying by the reciprocal of this fraction. This means the tan of seven 𝜋 over six is equal to negative one-half multiplied by negative two over root three. Cross canceling by a factor of two and then multiplying the numerators and denominators separately, we have one over root three.
Finally, we can rationalize the denominator by multiplying both the numerator and denominator by root three. The exact value of the tan of seven 𝜋 over six is root three over three.
