# Right Angle Trigonometry

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“Trigonometry” means “measuring triangles.” Basic trigonometry refers to the relationships between the measures of angles and sides of right triangles on a flat plane.

You have already seen some of these relationships in our old friend the 3-4-5 right triangle.

Notice the Greek letter *theta* θ in the upper angle of the diagram. It doesn’t mean anything in particular. Math people use it because it looks cool. It refers to a trigonometry function.

## What is a Trig Function?

The acute angles of a right triangle can be expressed either in degrees or as a ratio of side lengths. For example, angle θ in the diagram above measures approximately 53°. But it can be expressed more precisely as a ratio of the opposite side to the hypotenuse or 4/5. This ratio is called the sine function. The three basic trig functions are sine, cosine, and tangent.

sine = ^{opposite}/_{hypotenuse}

The * sine* of angle θ is the ratio of the

*side over the*

**opposite***, or 4 over 5.*

**hypotenuse**cosine = ^{adjacent}/_{hypotenuse}

The * cosine* of angle θ is the ratio of the

*side over the*

**adjacent***or*

**hypotenuse**^{3}/

_{5}.

tangent = ^{opposite}/_{adjacent}

The * tangent* of angle θ is the ratio of the

*side over the*

**opposite***, or*

**adjacent**^{4}/

_{3}.

## SOHCAHTOA

If you take the first letter of each of the bold-face words in the previous three sentences, you get the following word: **sohcahtoa**. This word can help you memorize the three basic trigonometry functions.

You will often find *sine* abbreviated to *sin*, *cosine* abbreviated to *cos*, and *tangent* abbreviated to *tan*.

## Reciprocals of Trig Functions

“Reciprocal” is a fancy word for flipping a fraction. Since the basic trig functions are ratios (fractions), they can be flipped upside down to make their reciprocals.

The reciprocal of sine is cosecant (csc): | csc = | ^{1}/_{sin} |

The reciprocal of cosine is secant (sec): | sec = | ^{1}/_{cos} |

The reciprocal of tangent is cotangent (cot): | cot = | ^{1}/_{tan} |

Consider the angle θ of the 3-4-5 triangle again.

If sin θ = ^{4}/_{5}, then csc θ = ^{5}/_{4}.

If cos θ = ^{3}/_{5}, then sec θ = ^{5}/_{3}.

If tan θ = ^{4}/_{3}, then cot θ = ^{3}/_{4}.