Trigonometric functions

The trigonometric functions, often shortened in everyday speech to trig functions, are a series of real functions in trigonometry identifying an angle in a right-angled triangle to two of its side lengths.

The sides of a triangle

The three sides of a triangle are labelled differently depending on the angle or side on which you are focused:

• The hypotenuse is the longest side
• The adjacent is adjacent (next to) the unknown angle, which we call θ ("theta").
• The opposite is the side opposite θ.

The six functions

The three main functions are expressed as such that for any angle size θ:

• Sine: sinθ = Opposite / Hypotenuse
• Cosine: cosθ = Adjacent / Hypotenuse
• Tangent: tanθ = Opposite / Adjacent

Students are traditionally asked to remember the main trigonometric functions using the initialism SOHCAHTOA.

There are three additional functions:

• Cosecant: cscθ = Hypotenuse / Opposite
• Secant: secθ = Hypotenuse / Adjacent
• Cotangent: cotθ = Adjacent / Opposite

The six functions also relate to each other as thus:

$\tan(x) = \sin(x) \div \cos(x)$

$\sec(x) = 1 \div \cos(x)$

$\cot(x) = \cos(x) \div \sin(x) = 1 \div \tan(x)$

$\csc(x) = 1 \div \sin(x)$

More regarding the application of the six functions are included in trigonometric identities.

Taylor series

$\sin(x)=x^1\div(1!)-x^3\div(3!)+x^5\div(5!)-x^7\div(7!)+\cdots$ $\cos(x)=x^0\div(0!)-x^2\div(2!)+x^4\div(4!)-x^6\div(6!)+\cdots$

Sine and cosine rules

Two of the greater rules, sine and cosine, have their own rules attached to them.

• The sine rule is that when finding a side, $\sin(a) \div a = \sin(b) \div b = \sin(c) \div c$
  • When finding an angle, the rule is flipped over: $a \div \sin(a) = b \div \sin(b) = c \div \sin(c)$
• The cosine rule for finding a side is that $a^2 = b^2 + c^2 - 2bc\cos(A)$
  • When finding an angle it is rearranged: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://mathoid-facade/v1/":): {\displaystyle \cos(A) = (b^2 + c^2 - a^2) \div 2bc}

Inverse sine, cosine and tangent

Where a triangle requires you to find an angle instead of a side with sine, we use the inverse sine (occasionally called the arcsine), expressed as sin^-1, or $\arcsin$.

Similarly, for cosine we use the inverse cosine, or the acos (cos^-1 or $\arccos$); and for tangent we use the inverse tangent, or the atan (tan^-1 or $\arctan$).

However, inverse sine, inverse cosine and inverse tangent will only ever return one answer. On a sine graph, since 360 degrees are added along the x-axis, there are infinitely many answers.