unit circle again
  • The position of every point in the Cartesian plane can be described as a scalar times a direction vector. A direction vector has unit length.
  • Every Complex number can be represented as the product of a real number and a complex number with modulus \(1\).
  • Every Complex number can be represented as the product of a real number and a complex number that sits on the unit circle.

To understand the Complex numbers, we just need to understand the unit circle. To understand the unit circle, we just need to understand right triangles.

Similar Triangles

\(\blacktriangleright \) Similar Triangles

From Geometry, we know that triangles with the same three angles are called similar. Similar triangles can be of different sizes, but their sides are always proportional.

\(\blacktriangleright \) Unit Circle

If we pick \(\theta \), then we have defined a right triangle via the unit circle.

The hypotenuse is the radius of the circle, which has a length of \(1\).

By extending the radius to larger circles we can create many right triangles. As long as the angle stays the same, then we have similar triangles and all of the sides are proportional.

We can use this to extend our right triangles on the unit circle outward into the Complex Plane.

Extending Right Triangles

Since our initial right triangle was anchored to the unit circle, its hypotenuse has length \(1\). The hypotenuses of the other similar right triangles have a length, which we’ll call \(r\).

The two right triangles are similar.

The ratio of corresponding sides are equal.

This has to be true for any right triangle with an angle \(\theta \).

If we label the sides relative to the angle \(\theta \), then we obtain descriptive ratios.

The ratios have to be the same as for the unit circle triangle.

\[ \sin (\theta ) = \frac {opp}{hyp} \]
\[ \cos (\theta ) = \frac {adj}{hyp} \]
\[ \tan (\theta ) = \frac {\sin (\theta )}{\cos (\theta )} = \frac {opp}{adj} \]

Other Quadrants

Everything holds for larger angles. We just have to bridge two thoughts.

  • The coordinates of points might be negative.
  • Lengths of sides of triangles are not negative.

To help bridge these two thoughts we will have two angles and two triangles.

  • The actual angle, \(\theta \), rotating from the positive \(x\)-axis.
  • The reference angle, \(\phi \), which will sit inside our reference triangle formed with the horizontal axis.

Once we rotate out of the first quadrant, then we need to switch our visual right triangle.

If \(\theta \) is in the second quadrant, then we use a new right triangle formed with the \(x\)-axis. \(\theta \) sweeps counterclockwise from the positive \(x\)-axis out to our hypotenuse. However, \(\theta \) is no longer inside our reference right triangle. Instead we have the angle \(\phi = \pi - \theta \).

The point at the end of our hypotenuse has coordinates, \((r \cos (\theta ), r \sin (\theta ))\). In the second quadrant, these coordinates are negative and positive: \(r \cos (\theta ) < 0\) and \(r \sin (\theta ) > 0\).

However, practically speaking, we use the reference angle and reference triangle to calculate values. After calculating the lengths of the sides of the reference triangle, we must remember to change their signs appropriately.

We can easily extend this example to quadrants III and IV.

  • \(\cos \left ( \frac {5 \pi }{4} \right ) = -\cos \left ( \frac {\pi }{4} \right ) = -\frac {1}{\sqrt {2}}\)
  • \(\sin \left ( \frac {5 \pi }{4} \right ) = -\sin \left ( \frac {\pi }{4} \right ) = -\frac {1}{\sqrt {2}}\)
  • \(\cos \left ( \frac {7 \pi }{4} \right ) = \cos \left ( \frac {\pi }{4} \right ) = \frac {1}{\sqrt {2}}\)
  • \(\sin \left ( \frac {7 \pi }{4} \right ) = -\sin \left ( \frac {\pi }{4} \right ) = -\frac {1}{\sqrt {2}}\)
Signs

In the first quadrant, sine is negative positive.

In the second quadrant, sine is negative positive.

In the third quadrant, sine is negative positive.

In the fourth quadrant, sine is negative positive.

Signs

In the first quadrant, cosine is negative positive.

In the second quadrant, cosine is negative positive.

In the third quadrant, cosine is negative positive.

In the fourth quadrant, cosine is negative positive.

Calculator

Your calculator has buttons for sine, cosine, and tangent. You supply an angle and the calculator gives you back the trigonometric ratio in decimal form.

The calculator can only give approximations, but for most applications, that is enough.

Note: Your calculator has two modes for angle measurements: degrees and radians. Remember to switch between these modes when calculating.

Calculator

Approximate the following.

  • \(\cos (35^{\circ }) \approx \answer [tolerance=0.001]{0.8191520443}\)
  • \(\sin (155^{\circ }) \approx \answer [tolerance=0.001]{0.4226182617}\)
  • \(\cos \left ( \frac {9\pi }{7} \right ) \approx \answer [tolerance=0.001]{-0.6234898016}\)
  • \(\sin \left ( \frac {9\pi }{5} \right ) \approx \answer [tolerance=0.001]{-0.5877852523}\)

Your calculator goes the other way as well.

You can also supply the calculator with the ratio and the calculator returns the angle.

  • If you have the sine ratio then the \(SIN^{-1}\) will return the angle. The angle will be in the first or fourth quadrant.
  • If you have the cosine ratio then the \(COS^{-1}\) will return the angle. The angle will be in the first or second quadrant.
  • If you have the tangent ratio then the \(TAN^{-1}\) will return the angle. The angle will be in the first or fourth quadrant.

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more examples can be found by following this link
More Examples of Right Triangles

2025-09-20 21:00:33