AS/A2 Pure 2 Pure 3

P1 Topic 3: Trigonometry
Unit circle definitions 1
Study the diagram below carefully. It shows a unit circle (in other words a circle whose radius is 1 unit) and a point on it that can be moved around that circle. Move the point. The angle formed by the line from the circle's centre to this point and the positive x axis is shown to the right. The distance indicated by the red arrow on the vertical scale shows the value of the sine function of the angle shown. You should be able to check that the sine of 90° is 1. Check also that sine(180°) = 0 and sine(270°) = -1 The orange arrow indicates the cosine of the angle shown. You should be able to check that the cosine of 90° is 0. Check also that cosine(180°) = -1 and cosine(270°) = 0. The blue arrow indicates a position on the vertical tangent line touching the circle to the right. This vertical distance gives the value for the tangent function of the angle shown. Check that the tangent of 0° is zero, and that tangent(45°) = 1. Check that tangent(135°) and tangent(315°) are -1 and also that tangent(90°) and tangent(270°) are undefined (this is more precise than to say infinite).

Please enable Java for an interactive construction (with Cinderella).

This display shows more simply the connection between the angle and the values of the three trignometrical functions. Drag the green point.

Summary The sine, cosine and tangent functions are valid for all angles from 0° to 360°, and beyond. The sine and cosine functions have values between -1 and 1. The tangent function can take all values. Cinderella interactive geometry. Trigonometry applets from catcode.com