 ## Trigonometry

Trigonometry is a branch of mathematics involving the study of triangles. Ancient builders and mariners used it for finding lengths that are not physically measurable (because they were so large) but they could be defined by angles.

Trigonometry has applications in fields such as engineering, surveying, navigation, optics, electronics, aviation, and cosmology in finding distances between stars and planets.

• T1 Pythagoras’ theorem

Pythagoras’ Theorem shows the relationship between the sides of a right-angled triangle. Knowing the length of two sides of a right-angled triangle, the length of the third side can be calculated. This mathematical formula is fundamental for finding lengths and distances that are difficult to physically measure.

• T2 Right triangle trigonometry

Sine, cos and tan can be defined using side lengths of a right-angled triangle. These side lengths are identified as either the hypotenuse or the opposite or adjacent sides to the angle. This module shows how to apply trigonometric ratios to find a missing side length, or angle, in a right-angled triangle.

• T3 The sine rule

How can we apply trigonometry to triangles that do not possess a right-angle? The sine rule shows that the ratio of the length of a side, to the sine of its opposite angle, will be the same for all three sides.

• T4 Cosine rule

The cosine rule is a generalisation of Pythagoras’ theorem. If you have any two sides of a triangle, as long as you know the angle between them, you can calculate the length of the third side.

• T5 Angular measurement and the unit circle

Angles are frequently measured in degrees. However, it is sometimes useful to define angles in terms of the length around the unit circle (a circle of radius = 1). This module introduces radians as a measure of angle.

• T6 Circular functions

How do I make sense of trigonometric values for angles larger than 90 degrees? This page introduces you to the unit circle and how you can map these angles onto the unit circle to clearly understand their value and (+/-) sign. See FG6 Circular functions

• T7 Trigonometric equations

If you know the value of a trigonometric function, how do I find all the possible angles that satisfy this expression? The calculator may only give you one answer to an inverse trig question between 0 and 90 degrees (say InvCos = 40°). The unit circle can help you visualise the many different solutions for finding your angle α.

• T8 Graphs of sine and cosine functions

Sine, cosine and tan functions can be graphed on x-y axes. Read this pdf to see how the amplitude (height) and period or frequency of these functions can be changed. See how the graph can be translated (shifted) along the horizontal or vertical axes. See FG10 Graphs of sine and cosine functions

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