| sin(theta) = a / c | csc(theta) = 1 / sin(theta) = c / a |
| cos(theta) = b / c | sec(theta) = 1 / cos(theta) = c / b |
| tan(theta) = sin(theta) / cos(theta) = a / b | cot(theta) = 1/ tan(theta) = b / a |
sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
| sin^2(x) + cos^2(x) = 1 | tan^2(x) + 1 = sec^2(x) | cot^2(x) + 1 = csc^2(x) | |
sin(x  y) = sin x cos y cos x sin y | |||
cos(x y) = cos x cosy  sin x sin y | |||
tan(x y) = (tan x tan y) / (1 tan x tan y)
y) = (tan x tan y) / (1 tan x tan y)
sin(2x) = 2 sin x cos x
cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)
tan(2x) = 2 tan(x) / (1 - tan^2(x))
sin^2(x) = 1/2 - 1/2 cos(2x)
cos^2(x) = 1/2 + 1/2 cos(2x)
sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )
cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )
| angle | 0 | 30 | 45 | 60 | 90 |
|---|---|---|---|---|---|
| sin^2(a) | 0/4 | 1/4 | 2/4 | 3/4 | 4/4 |
| cos^2(a) | 4/4 | 3/4 | 2/4 | 1/4 | 0/4 |
| tan^2(a) | 0/4 | 1/3 | 2/2 | 3/1 | 4/0 |
Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:
a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)
| (Law of Cosines) |
(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)
radians = degrees x PI / 180 (deg to rad conversion) degrees = radians x 180 / PI (rad to deg conversion)
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Hyperbolic Definitions
sinh(x) = ( e x - e -x )/2csch(x) = 1/sinh(x) = 2/( e x - e -x )cosh(x) = ( e x + e -x )/2
sech(x) = 1/cosh(x) = 2/( e x + e -x )
tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )
coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )
cosh 2(x) - sinh 2(x) = 1
tanh 2(x) + sech 2(x) = 1
coth 2(x) - csch 2(x) = 1
Inverse Hyperbolic Definitions
arcsinh(z) = ln( z +
(z 2 + 1) )arccosh(z) = ln( z (z 2 - 1) )arctanh(z) = 1/2 ln( (1+z)/(1-z) )
arccsch(z) = ln( (1+
(1+z 2) )/z )arcsech(z) = ln( (1
(1-z 2) )/z )arccoth(z) = 1/2 ln( (z+1)/(z-1) )
Relations to Trigonometric Functions
sinh(z) = -i sin(iz)csch(z) = i csc(iz)cosh(z) = cos(iz)
sech(z) = sec(iz)
tanh(z) = -i tan(iz)
coth(z) = i cot(iz)
| Overview |
| ( Math | Algebra | Function |Overview) |
Trig Functions: OverviewUnder its simplest definition, a trigonometric (literally, a "triangle-measuring") function, is one of the many functions that relate one non-right angle of a right triangle to the ratio of the lengths of any two sides of the triangle (or vice versa).
Any trigonometric function (f), therefore, always satisfies either of the following equations:
Under a certain convention, we label the sides as opposite, adjacent, and hypotenuse relative to our angle of interest q. full explanation
As mentioned previously, the first type of trigonometric function, which relates an angle to a side ratio, always satisfies the following equation:
The three diagonal functions shown in red always equal one. They are degenerate and, therefore, are of no use to us. We therefore remove these degenerate functions and assign labels to the remaining six, usually written in the following order:
Furthermore, the functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
Do not be overwhelmed. By far, the two most important trig functions to remember are sine and cosine. All the other trig functions of the first kind can be derived from these two functions. For example, the functions on the right are merely the multiplicative inverse of the corresponding function on the left (that makes them much less useful). Furthermore, the sin(x) / COs(x) = (opp/hyp) / (adj/hyp) = opp / adj = tan(x). Therefore, the tangent function is the same as the quotient of the sine and cosine functions (the tangent function is still fairly handy).
Let's examine these functions further. You will notice that there are the sine, secant, and tangent functions, and there are corresponding "co"-functions. They get their odd names from various similar ideas in geometry. You may suggest that the cofunctions should be relabeled to be the multiplicative inverses of the corresponding sine, secant, and tangent functions. However, there is a method to this madness. Acofunction of a given trig function (f) is, by definition, the function obtained after the complement its parameter is taken. Since the complement of any angle q is 90° - q, the the fact that the following relations can be shown to hold:
sine(90° - q) = cosine(q)
Just as we can define trigonometric functions of the form
As before, the functions are usually abbreviated: arcsine (arcsin), arccosine (arccos), arctangent (arctan) arccosecant (arccsc), arcsecant (arcsec), and arccotangent (arccot). According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 csc-1, sec-1, and cot-1. Beware: There is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).
Any trigonometric function (f), therefore, always satisfies either of the following equations:
f(q) = a / b OR f(a / b) = q,
where q is the measure of a certain angle in the triangle, and a and b are the lengths of two specific sides.This means that- If the former equation holds, we can choose any right triangle, then take the measurement of one of the non-right angles, and when we evaluate the trigonometric function at that angle, the result will be the ratio of the lengths of two of the triangle's sides.
- However, if the latter equation holds, we can chose any right triangle, then compute the ratio of the lengths of two specific sides, and when we evaluate the trigonometric function at any that ratio, the result will be measure of one of the triangles non-right angles. (These are called inverse trig functions since they do the inverse, or vice-versa, of the previous trig functions.)
Under a certain convention, we label the sides as opposite, adjacent, and hypotenuse relative to our angle of interest q. full explanation
As mentioned previously, the first type of trigonometric function, which relates an angle to a side ratio, always satisfies the following equation:
f(q) = a / b.
Since given any angle q, there are three ways of choosing the numerator (a), and three ways of choosing the denominator (b), we can create the following nine trigonometric functions:| f(q) = opp/opp | f(q) = opp/adj | f(q) = opp/hyp |
| f(q) = adj/opp | f(q) = adj/adj | f(q) = adj/hyp |
| f(q) = hyp/opp | f(q) = hyp/adj | f(q) = hyp/hyp |
The three diagonal functions shown in red always equal one. They are degenerate and, therefore, are of no use to us. We therefore remove these degenerate functions and assign labels to the remaining six, usually written in the following order:
| sine(q) = opp/hyp | cosecant(q) = hyp/opp |
| cosine(q) = adj/hyp | secant(q) = hyp/adj |
| tangent(q) = opp/adj | cotangent(q) = adj/opp |
Furthermore, the functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
Do not be overwhelmed. By far, the two most important trig functions to remember are sine and cosine. All the other trig functions of the first kind can be derived from these two functions. For example, the functions on the right are merely the multiplicative inverse of the corresponding function on the left (that makes them much less useful). Furthermore, the sin(x) / COs(x) = (opp/hyp) / (adj/hyp) = opp / adj = tan(x). Therefore, the tangent function is the same as the quotient of the sine and cosine functions (the tangent function is still fairly handy).
| sine(q) = opp/hyp | CSC(q) = 1/sin(q) |
| COs(q) = adj/hyp | sec(q) = 1/COs(q) |
| tan(q) = sin(q)/COs(q) | cot(q) = 1/tan(q) |
Let's examine these functions further. You will notice that there are the sine, secant, and tangent functions, and there are corresponding "co"-functions. They get their odd names from various similar ideas in geometry. You may suggest that the cofunctions should be relabeled to be the multiplicative inverses of the corresponding sine, secant, and tangent functions. However, there is a method to this madness. Acofunction of a given trig function (f) is, by definition, the function obtained after the complement its parameter is taken. Since the complement of any angle q is 90° - q, the the fact that the following relations can be shown to hold:
sine(90° - q) = cosine(q)
secant(90° - q) = cosecant(q)
tangent(90° - q) = cotangent(q)
thus justifying the naming convention.The trig functions evaluate differently depending on the units on q, such as degrees, radians, or grads. For example, sin(90°) = 1, while sin(90)=0.89399.... explanationJust as we can define trigonometric functions of the form
f(q) = a / b
that take a non-right angle as its parameter and return the ratio of the lengths of two triangle sides, we can do the reverse: define trig functions of the form
f(a / b) = q
that take the ratio of the lengths of two sides as a parameter and returns the measurement of one of the non-right angles.| arcsine(opp/hyp) = q | arccosecant(hyp/opp) = q |
| arccosine(adj/hyp) = q | arcsecant(hyp/adj) = q |
| arctangent(opp/adj) = q | arccotangent(adj/opp) = q |
As before, the functions are usually abbreviated: arcsine (arcsin), arccosine (arccos), arctangent (arctan) arccosecant (arccsc), arcsecant (arcsec), and arccotangent (arccot). According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 csc-1, sec-1, and cot-1. Beware: There is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).
Trig Functions: The Functions
The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
It is often simpler to memorize the the trig functions in terms of only sine and cosine:
The functions are usually abbreviated:
arcsine (arcsin)
arccosine (arccos)
arctangent (arctan)
arccosecant (arccsc)
arcsecant (arcsec)
arccotangent (arccot).
According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 arccsc-1, arcsec-1, and arccot-1. Beware, though, there is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).
Trig Functions: Unit Modes
| sine(q) = opp/hyp | cosecant(q) = hyp/opp |
| cosine(q) = adj/hyp | secant(q) = hyp/adj |
| tangent(q) = opp/adj | cotangent(q) = adj/opp |
The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
It is often simpler to memorize the the trig functions in terms of only sine and cosine:
| sine(q) = opp/hyp | csc(q) = 1/sin(q) |
| cos(q) = adj/hyp | sec(q) = 1/COs(q) |
| tan(q) = sin(q)/cos(q) | cot(q) = 1/tan(q) |
| arcsine(opp/hyp) = q | arccosecant(hyp/opp) = q |
| arccosine(adj/hyp) = q | arcsecant(hyp/adj) = q |
| arctangent(opp/adj) = q | arccotangent(adj/opp) = q |
The functions are usually abbreviated:
arcsine (arcsin)
arccosine (arccos)
arctangent (arctan)
arccosecant (arccsc)
arcsecant (arcsec)
arccotangent (arccot).
According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 arccsc-1, arcsec-1, and arccot-1. Beware, though, there is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).
Trig Functions: Unit Modes
The trig functions evaluate differently depending on the units on q. For example, sin(90°) = 1, while sin(90)=0.89399.... If there is a degree sign after the angle, the trig function evaluates its parameter as a degree measurement. If there is no unit after the angle, the trig function evaluates its parameter as a radian measurement. This is because radian measurements are considered to be the "natural" measurements for angles. (Calculus gives us a justification for this. A partial explanation comes from the formula for the area of a circle sector, which is simplest when the angle is in radians).
Calculator note: Many calculators have degree, radian, and grad modes (360° = 2p rad = 400 grad). It is important to have the calculator in the right mode since that mode setting tells the calculator which units to assume for angles when evaluating any of the trigonometric functions. For example, if the calculator is in degree mode, evaluating sine of 90 results in 1. However, the calculator returns 0.89399... when in radian mode. Having the calculator in the wrong mode is a common mistake for beginners, especially those that are only familiar with degree angle measurements.
For those who wish to reconcile the various trig functions that depend on the units used, we can define the degree symbol (°) to be the value (PI/180). Therefore, sin(90°), for example, is really just an expression for the sine of a radian measurement when the parameter is fully evaluated. As a demonstration, sin(90°) = sin(90(PI/180)) = sin(PI/2). In this way, we only need to tabulate the "natural" radian version of the sine function. (This method is similar to defining percent % = (1/100) in order to relate percents to ratios, such as 50% = 50(1/100) = 1/2.)
Trig: Labeling Sides
Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.
We need a way of labeling the sides. Consider a general right triangle:
A right triangle has two non-right angles, and we choose one of these angles to be our angle of interest, which we label "q." ("q" is the Greek letter "theta.")
We can then uniquely label the three sides of the right triangle relative to our choice of q. As the above picture illustrates, our choice of q affects how the three sides get labeled.
We label the three sides in this manner: The side opposite the right angle is called the hypotenuse. This side is labeled the same regardless of our choice of q. The labeling of the remaining two sides depend on our choice of theta; we therefore speak of these other two sides as beingadjacent to the angle q or opposite to the angle q. The remaining side that touches the angle q is considered to be the side adjacent to q, and the remaining side that is far away from the angle q is considered to beopposite to the angle q, as shown in the picture.
Calculator note: Many calculators have degree, radian, and grad modes (360° = 2p rad = 400 grad). It is important to have the calculator in the right mode since that mode setting tells the calculator which units to assume for angles when evaluating any of the trigonometric functions. For example, if the calculator is in degree mode, evaluating sine of 90 results in 1. However, the calculator returns 0.89399... when in radian mode. Having the calculator in the wrong mode is a common mistake for beginners, especially those that are only familiar with degree angle measurements.
For those who wish to reconcile the various trig functions that depend on the units used, we can define the degree symbol (°) to be the value (PI/180). Therefore, sin(90°), for example, is really just an expression for the sine of a radian measurement when the parameter is fully evaluated. As a demonstration, sin(90°) = sin(90(PI/180)) = sin(PI/2). In this way, we only need to tabulate the "natural" radian version of the sine function. (This method is similar to defining percent % = (1/100) in order to relate percents to ratios, such as 50% = 50(1/100) = 1/2.)
Trig: Labeling Sides
Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.
We need a way of labeling the sides. Consider a general right triangle:
A right triangle has two non-right angles, and we choose one of these angles to be our angle of interest, which we label "q." ("q" is the Greek letter "theta.")
We can then uniquely label the three sides of the right triangle relative to our choice of q. As the above picture illustrates, our choice of q affects how the three sides get labeled.
We label the three sides in this manner: The side opposite the right angle is called the hypotenuse. This side is labeled the same regardless of our choice of q. The labeling of the remaining two sides depend on our choice of theta; we therefore speak of these other two sides as beingadjacent to the angle q or opposite to the angle q. The remaining side that touches the angle q is considered to be the side adjacent to q, and the remaining side that is far away from the angle q is considered to beopposite to the angle q, as shown in the picture.
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