Question

Use the unit circle to find tan 30°

Answer 1

Answer:

A

Step-by-step explanation:

Draw the unit circle. Now draw the vertical line (the tangent line) that passes through the point (1,0). In anticlockwise direction from the point (1,0) draw the line that makes 30° angle with positive direction of x axis. Consider right triangle formed by the circle and these two lines. In this triangle the horizontal leg has length 1 (because this leg is the radius of the unit circle). The vertical leg has the length x. The leg opposite to the 30° angle is half of the hypotenuse, so the hypotenuse is 2x. By the Pythagorean theorem,

$(2x)^2=1^2+x^2,\\ \\4x^2=1+x^2,\\ \\3x^2=1,\\ \\x^2=\dfrac{1}{3},\\ \\x=\sqrt{\dfrac{1}{3}}=\dfrac{\sqrt{3}}{3}.$

By the definition of tangent,

$\tan 30^{\circ}=\dfrac{\text{opposite leg}}{\text{adjacent leg}}=\dfrac{\frac{\sqrt{3}}{3}}{1}=\dfrac{\sqrt{3}}{3}.$