Unit Circle

Use the animation by dragging the red dot around the edge of the circle. As the dot moves, the angle \( \theta \) changes, and the display underneath shows the size of that angle in both degrees and radians. Watch the point’s coordinates as it moves: the \(x\)-coordinate gives \( \cos \theta \), and the \(y\)-coordinate gives \( \sin \theta \). This helps you see that the sine and cosine ratios are linked directly to positions on a circle of radius 1.

The animation also shows how the tangent ratio changes. When the dot is near the top or bottom of the circle, the tangent value becomes very large, because \( \cos \theta \) is close to zero and \( \tan \theta = \dfrac{\sin \theta}{\cos \theta} \). You can use the animation to spot patterns, such as which angles have positive or negative sine, cosine and tangent values, why some angles have the same sine or cosine value, and how special angles like \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\) and \(180^\circ\) appear in radians.

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Trig Tour

See angles, in either degrees or radians, on the unit circle and the graph of the trigonometric function.

The short web address is:

Transum.org/go/?Num=1061

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Trigonometry

Sine, cosine and tangent ratios are used to find sides and angles in right-angled triangles.

The short web address is:

Transum.org/go/?Num=71

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If Then Trigonometry

Finding the exact values of sine, cosine and tangent of angles if given a different trig ratio.

The short web address is:

Transum.org/go/?Num=896

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Common Trig Ratios Degrees

A self-marking exercise on finding the exact values of sine, cosine and tangent of special angles.

The short web address is:

Transum.org/go/?Num=543

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Common Trig Ratios Radians

A self-marking exercise on finding the exact values of sine, cosine and tangent of special angles given in radians.

The short web address is:

Transum.org/go/?Num=1054

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Trigonometry Advanced

A self-marking exercise on the sine rule, cosine rule and the sine formula for finding the area of a triangle.

The short web address is:

Transum.org/go/?Num=662

Here are some basic, general questions, that you might like to investigate.

1. What are the values of \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \) when \( \theta = 0^\circ \)?
2. At which angle between \(0^\circ\) and \(360^\circ\) does \( \sin \theta \) first reach its maximum value?
3. What are the coordinates of the point on the unit circle when \( \theta = 90^\circ \), and how do these coordinates relate to \( \cos \theta \) and \( \sin \theta \)?
4. Why is \( \tan 90^\circ \) undefined?
5. For which angles between \(0^\circ\) and \(360^\circ\) is \( \cos \theta = 0 \)?
6. What happens to the value of \( \tan \theta \) as the angle gets closer to \(90^\circ\) from the left?
7. In which quadrants is \( \sin \theta \) positive?
8. In which quadrants is \( \cos \theta \) negative?
9. Find two different angles between \(0^\circ\) and \(360^\circ\) that have the same sine value.
10. What is the radian measure of \(180^\circ\), and where is the point on the unit circle at this angle?

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