Trigonometry
Trigonometry is often visualized with a circle of radius 1 (the unit circle), and most trigonometry students are familar with the cosine and sine of an angle from the positive x-axis as the x and y coordinates of the point on the circle intersected by the other ray of the angle. The cosine and sine are often visualized as the legs of the right triangle defined by the origin, the point of intersection, and the point on the x-axis under the intersection point.
If you are more familiar with trigonometry without the unit circle (SOH CAH TOA) the simplicity of the unit circle might be a bit confusing. The cosine is adjacent/hypotenuse, but in this picture the cosine is pictured as just the adjacent. That's part of the beauty of the unit circle: in this triangle, the hypotenuse is 1 so
adjacent/hypotenuse = adjacent/1 = adjacent
Less familiar are the two triangles similar to this one which help visualize the other basic trig functions: tangent, secant, cotangent and cosecant.
The visualization looks intimidating at first but it's not hard to work out all the relationships, remembering or noticing:
- The definitions of these other functions in terms of the cosine and sine (e.g. tanθ = sinθ/cosθ)
- Similar triangles have corresponding sides in proportion and
- One of the sides of each triangle is a radius, so it has length 1.
So for example, to see why the vertical leg in the red triangle is labeled tanθ, notice that the horizontal leg of the red triangle has length 1. So in the red triangle, vertical/horizontal = vertical/1 = vertical. But since that triangle is similar to the black triangle, that ratio should also equal the black vertical/horizontal ratio, which is sinθ/cosθ = tanθ. The other sides of the red and blue triangles can be discovered similarly.
Can you see:
- Why the sides labeled secθ, cscθ, and cotθ deserve those labels (see the justification of tanθ above);
- Where the names tangent, cotangent, secant and cosecant come from? (Think about the geometric meanings of the words tangent, secant and complement);
- Why the three Pythagorean trig identities are true?
Try it! Press the "Open Geogebra" button; then drag the green intersection point to change the angle and see what happens.
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