Trigonometry

Trigonometry is the study of angles, triangles, and trigonometric functions such as sine, cosine and tangent. First used by Egyptians and Babylonians, trig has been around for over 3000 years. Trigonometry is used in surveying, navigation, and various sciences such as physics. For an excellent review of Trigonometry, trig subjects and its history, check out Wikipedia or Wolfram Math.

The Big Ideas covered in Trigonometry include:

Other topics that might help:

Right Triangle Trig

Most students begin their study of Trigonometry with right triangles. If you have a right triangle, you can name the angles with capital letters - A B, and C. By convention, angle C is always the right or 90 angle. The sides of the triangle are named using small letters and the letter matches the letter of the angle opposite or across from the side. The longest side c is also know as the hypotenuse.

Pythagoras is given credit for discovering what is now known as the Pythagorean Theorem in the 6th century BC. His theorem says that the three sides of every right triangle are related such that:

a2 + b2 = c2

If you know any two of the sides in a right triangle, you can find the dimensions for the missing side using the Pythagorean Theorem.

If you know that you have a right triangle and you know one of the other angles, you can find the missing angle. This is possible because we know that the three angles of a triangle always add to 180 and that angle C is 90 in a right triangle. From this, we get the following relationships:

A + B + C = 180
A + B = 90

Similar triangles are triangles with the same sized angles, but different sized sides. It was noticed early on that for similar triangles the ratios of the sides was a constant. These ratios are called the basic Trig Functions; sine, cosine, tangent, secant, cosecant, and cotangent. Their definitions are:

Trig Function Abbreviation

Definition

Sine sin
sin A = 

side opposite A

 = 

opp

hypotenuse hyp
Cosine cos
cos A = 

side adjacent to A

 = 

adj

hypotenuse hyp
Tangent tan
tan A = 

side opposite A

 = 

opp

side adjacent to A adj
Cosecant csc
csc A = 

hypotenuse

 = 

hyp

side opposite A opp
Secant sec
sec A = 

hypotenuse

 = 

hyp

side adjacent to A adj
Cotangent cot
cot A = 

side adjacent to A

 = 

adj

side opposite A opp

You can remember the pattern for sine, cosine and tangent as Old Hippies Are High On Acid where the first letter represent the first letters of the definition. The last three, cosecant, secant, and cotangent are just the reciprocals of the first three.


Unit Circles and trig for any angles

Once people knew all about angles and ratios in a right triangle, they discovered that you can also define the trig functions based on a circle with a radius of one, called a unit circle. I will add these sections later - check back soon!


Oblique Triangle Trig

Sometimes things do not form nice right triangles, so the rules for trig have been generalized to help you solve problems even when you don't have a right triangle.


Applications of Trig

Trigonometry was one of the oldest branches of applied mathematics. It was used by the ancient Greeks to find out distances that could not be measured directly. In fact, one ancient mathematician calculated the distance from the earth to the moon, and he got an answer that was surprisingly accurate for a time before calculators.


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Last Updated: 1/6/2011
K. A. Shearer 2005-2011
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