;[CALHELP2.CAL] This is XCAL's help for Trigonometric Native Functions ; ;HelpPage ;HelpSection Native Trigonometric Functions ;HelpStart FUNCTION,FUNCTIONS,NATIVE,TRIG,TRIGONOMETRIC ;HelpStart SIN,SIN(X),SIN2,SIN2(X),SIN3,SIN3(X) SIN(x) ------ The SIN function returns the dimensionless quantity Sine of x. If x is large, a partial loss of significance may occur and SIN generates a PLOSS error. If x is so large that significance is completely lost, a TLOSS error is generated. The value x may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. SIN2(x), read as sine squared of x, is simply SIN(x)^2. SIN3(x), read as sine cubed of x, is simply SIN(x)^3. In the formula y=SIN(x), the DOMAIN x is -Infinity..+Infinity while the RANGE y is -1..+1. Graphically, y=SIN(x) may be shown as follows: y +1+ * y=SIN(x) * | * * * * | * * * * | * * * * 0 |* * Pi 2*Pi * Radians * x ------+--------------------- * -------------------- * ---------------------- *|0 180 * * 360 Degrees * | * * * | * * * | * * -1+ * In a right triangle: SIN(x) = opposite / hypotenuse ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with SIN: SIN(x) = TAN(x)*COS(x) = 1/CSC(x) SIN(-x) = -SIN(x) SIN(2*x) = 2*SIN(x)*COS(x) SIN(3*x) = 3*SIN(x)-4*SIN(x)^3 SIN(4*x) = 4*SIN(x)*COS(x)-8*SIN(x)^3*COS(x) SIN(5*x) = 5*SIN(x)-20*SIN(x)^3+16*SIN(x)^5 SIN(x)^2 = 1-COS(x)^2 = COS(x)^2-COS(2*x) = 1/2-1/2*COS(x)^2 SIN(x)^3 = 3/4*SIN(x)-1/4*SIN(3*x) SIN(x)^4 = 3/8-1/2*COS(2*x)+1/8*COS(4*x) SIN(x)^5 = 5/8*SIN(x)-5/16*SIN(3*x)+1/16*SIN(5*x) SIN(a+b) = SIN(a)*COS(b)+COS(a)*SIN(b) SIN(a-b) = SIN(a)*COS(b)-COS(a)*SIN(b) SIN(a)+SIN(b) = 2*SIN(a+b)^(1/2)*COS(a-b)^(1/2) SIN(a)-SIN(b) = 2*COS(a+b)^(1/2)*SIN(a-b)^(1/2) SIN(a)*SIN(b) = 1/2*(COS(a-b)-COS(a+b)) SIN(a)*COS(b) = 1/2*(SIN(a-b)+COS(a+b)) ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd SIN,SIN(X),SIN2,SIN2(X),SIN3,SIN3(X) ;HelpPage ;HelpStart COS,COS(X),COS2,COS2(X),COS3,COS3(X) COS(x) ------ The COS function returns the dimensionless quantity Cosine of x. If x is large, a partial loss of significance may occur and COS generates a PLOSS error. If x is so large that significance is completely lost, a TLOSS error is generated. The value x may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. COS2(x), read as cosine squared of x, is simply COS(x)^2. COS3(x), read as cosine cubed of x, is simply COS(x)^3. In the formula y=COS(x), the DOMAIN x is -Infinity..+Infinity while the RANGE y is -1..+1. Graphically, y=COS(x) may be shown as follows: y +1 * y=COS(x) * * | * * * * | * * * * | * * * * -Pi/2 |0 * Pi/2 3/2*Pi * Radians * x * ----------+--------- * -------------------- * ---------------------- * --- * -90 0| 90 * * 270 Degrees * | * * * | * * * | * * -1 + * In a right triangle: COS(x) = adjacent / hypotenuse ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with COS: COS(x) = SIN(x)/TAN(x) = 1/SEC(x) COS(-x) = COS(x) COS(2*x) = COS(x)^2-SIN(x)^2 = 1-2*SIN(x)^2 = 2*COS(x)^2-1 COS(3*x) = 4*COS(x)^3-3*COS(x) COS(4*x) = 8*COS(x)^4-8*COS(x)^2+1 COS(5*x) = 16*COS(x)^5-20*COS(x)^3+5*COS(x) COS(x)^2 = 1-SIN(x)^2 = SIN(x)^2+COS(2*x) = 1/2+1/2*COS(2*x) COS(x)^3 = 3/4*COS(x)+1/4*COS(3*x) COS(x)^4 = 3/8+1/2*COS(2*x)+1/8*COS(4*x) COS(x)^5 = 5/8*COS(x)+5/16*COS(3*x)+1/16*COS(5*x) COS(a+b) = COS(a)*COS(b)-SIN(a)*SIN(b) COS(a-b) = COS(a)*COS(b)+SIN(a)*SIN(b) COS(a)+COS(b) = 2*COS(a+b)^(1/2)*COS(a-b)^(1/2) COS(a)-COS(b) = 2*SIN(a+b)^(1/2)*SIN(b-a)^(1/2) COS(a)*COS(b) = 1/2*(COS(a-b)+COS(a+b)) COS(a)*SIN(b) = 1/2*(SIN(a-b)+COS(a+b)) ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd COS,COS(X),COS2,COS2(X),COS3,COS3(X) ;HelpPage ;HelpStart TAN,TAN(X),TAN2,TAN2(X),TAN3,TAN3(X) TAN(x) ------ The TAN function returns the dimensionless quantity Tangent of x. If x is large, a partial loss of significance may occur and TAN generates a PLOSS error. If x is so large that significance is completely lost, a TLOSS error is generated. The value x may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. TAN2(x), read as tangent squared of x, is simply TAN(x)^2. TAN3(x), read as tangent cubed of x, is simply TAN(x)^3. In the formula y=TAN(x), both the DOMAIN and the RANGE of TAN() is -Infinity to +Infinity. Graphically, y=TAN(x) may be shown as follows: +Infinity y *| | *| | *| y=TAN(x) *| | *| | *| * | | * | | * | * |-Pi/2 |0 * |Pi/2 |Pi * |3*Pi/2 Radians * | | * | | * | * | | * | | * | x ---------+---------- + ----------+---------- * ----------+----------- x | * | | * | | * | * | | * | | * -90| * 0| 90| * 180| 270| * Degrees | * | | * | | * | * | | * | | * |* | |* | |* |* | |* | |* -Infinity y In a right triangle: TAN(x) = opposite / adjacent ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with TAN: TAN(x) = SIN(x)/COS(x) = 1/COT(x) TAN(-x) = -TAN(x) TAN(x)^2 = 1-SEC(x)^2 TAN(2*x) = 2*TAN(x)/(1-TAN(x)^2) TAN(3*x) = (3*TAN(x)-TAN(x)^3)/(1-3*TAN(x)^2) TAN(4*x) = (4*TAN(x)-4*TAN(x)^3)/(1-6*TAN(x)^2+TAN(x)^4) TAN(5*x) = (TAN(x)^5-10*TAN(x)^3+5*TAN(x))/(1-10*TAN(x)^2+5*TAN(x)^4) TAN(a+b) = (TAN(a)+TAN(b))/(1-TAN(a)*TAN(b)) TAN(a-b) = (TAN(a)-TAN(b))/(1+TAN(a)*TAN(b)) ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd TAN,TAN(X),TAN2,TAN2(X),TAN3,TAN3(X) ;HelpPage ;HelpStart COT,COT(X),COT2,COT2(X),COT3,COT3(X) COT(x) ------ The COT function returns the dimensionless quantity CoTangent of x. If x is large, a partial loss of significance may occur and COT generates a PLOSS error. If x is so large that significance is completely lost, a TLOSS error is generated. The value x may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. COT2(x), read as cotangent squared of x, is simply COT(x)^2. COT3(x), read as cotangent cubed of x, is simply COT(x)^3. In the formula y=COT(x), both the DOMAIN and the RANGE of COT() are -Infinity to +Infinity. Graphically, y=COT(x) may be shown as follows: +Infinity y |* | |* | |* y=COT(x) |* | |* | |* | * | | * | | * | * | | * | | * 0| * 90| 180| * 270| 360| * Degrees | * | | * | | * | * | | * | | * x ---------+---------- * ----------+---------- * ----------+----------- x * | | * | | * | * | | * | | * | * |0 |Pi/2 * |Pi |3*Pi/2 * |2*Pi Radians * | | * | | * | *| | *| | *| *| | *| | *| -Infinity y In a right triangle: COT(x) = adjacent / opposite ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with COT: COT(x) = COS(x)/SIN(x) = 1/TAN(x) COT(-x) = -COT(x) COT(x)^2 = 1-CSC(x)^2 COT(a+b) = (COT(a)*COT(b)-1)/(COT(b)+COT(a)) COT(a-b) = (COT(a)*COT(b)+1)/(COT(b)-COT(a)) ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd COT,COT(X),COT2,COT2(X),COT3,COT3(X) ;HelpPage ;HelpStart SEC,SEC(X),SEC2,SEC2(X),SEC3,SEC3(X) SEC(x) ------ The SEC function returns the dimensionless quantity Secant of x. If x is large, a partial loss of significance may occur and SEC generates a PLOSS error. If x is so large that significance is completely lost, a TLOSS error is generated. The value x may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. SEC2(x), read as secant squared of x, is simply SEC(x)^2. SEC3(x), read as secant cubed of x, is simply SEC(x)^3. In the formula y=SEC(x), both the DOMAIN and the RANGE of SEC(x) are -Infinity..+Infinity. Graphically, y=SEC(x) may be shown as follows: y |* | *| y=SEC(x) |* | +Infinity | | | | | Radians | * | * | | * | -Pi/2| * 0| * |Pi/2 3*Pi/2| * | | * | * | | * | | * | * | | * | | * | | * +1 x-----------+----------+----------+---------------------+----------+ 0 * | | | * | | -1 * | | | * * | | * | | | * * | | Degrees * |-90 0| 90| * * |270 | * | | | * * | | | | | | | *| | |* *| | -Infinity In a right triangle: SEC(x) = hypotenuse / adjacent ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with SEC: SEC(x) = TAN(x)/SIN(x) = 1/COS(x) SEC(-x) = SEC(x) SEC(x)^2 = 1-TAN(x)^2 ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd SEC,SEC(X),SEC2,SEC2(X),SEC3,SEC3(X) ;HelpPage ;HelpStart CSC,CSC(X),CSC2,CSC2(X),CSC3,CSC3(X) CSC(x) ------ The CSC function returns the dimensionless quantity Cosecant of x. If x is large, a partial loss of significance may occur and CSC generates a PLOSS error. If x is so large that significance is completely lost, a TLOSS error is generated. The value x may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. CSC2(x), read as cosecant squared of x, is simply CSC(x)^2. CSC3(x), read as cosecant cubed of x, is simply CSC(x)^3. In the formula y=CSC(x), both the DOMAIN and the RANGE of CSC(x) are -Infinity..+Infinity. Graphically, y=CSC(x) may be shown as follows: y |* *| y=CSC(x) |* | +Infinity | | | | | * * | | * | Radians 0| * * |Pi 2*pi| * | | * * | | * | | * * | | * | | * | | * +1 x-----------+---------------------+---------------------+----------+ 0 * | | * | | -1 * | | * * | | * | | * * | | Degrees * |0 180| * * |360 | * | | * * | | | | | | *| |* *| | -Infinity In a right triangle: CSC(x) = opposite / hypotenuse ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with CSC: CSC(x) = 1/TAN(x)*COS(x) = 1/SIN(x) CSC(-x) = -CSC(x) CSC(x)^2 = 1-COS(x)^2 ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd CSC,CSC(X),CSC2,CSC2(X),CSC3,CSC3(X) ***** ;HelpPage ;HelpStart ASIN,ASIN(X),ASIN2,ASIN2(X),ASIN3,ASIN3(X) ASIN(x) ------- The ASIN function accepts a dimensionless quantity and returns the angle Arc Sine of x. ASIN is the mathematical inverse of SIN (ie. x=ASIN(SIN(x)). The value y may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. ASIN2(x), read as Arc Sine squared of x, is simply ASIN(x)^2. ASIN3(x), read as Arc Sine cubed of x, is simply ASIN(x)^3. In the formula y=ASIN(x), the DOMAIN x is: -1 < x < 1. If the argument is outside this domain, ASIN returns zero and generates a "?DOMAIN error". The RANGE y is limited to the principal values of the Arc Sine of x (ie. -Pi/2 =< y =< Pi/2). ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with ASIN: ASIN(x) = Pi/2-ACOS(x) ASIN(-x) = -ASIN(x) ASIN(1/x) = ACSC(x) ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd ASIN,ASIN(X),ASIN2,ASIN2(X),ASIN3,ASIN3(X) ;HelpStart FUNCTION,FUNCTIONS,NATIVE,TRIG,TRIGONOMETRIC ;HelpEnd FUNCTION,FUNCTIONS,NATIVE,TRIG,TRIGONOMETRIC ;HelpStart ACOS,ACOS(X),ACOS2,ACOS2(X),ACOS3,ACOS3(X) ACOS(x) ------- The ACOS function accepts a dimensionless quantity and returns the angle Arc Cosine of x. ACOS is the mathematical inverse of COS (ie. x=ACOS(COS(x)). The value y may represent RADIANS or DEGREES and is interpreted as such by setting the /R or /NR command switches. ACOS2(x), read as Arc Cosine squared of x, is simply ACOS(x)^2. ACOS3(x), read as Arc Cosine cubed of x, is simply ACOS(x)^3. In the formula y=ACOS(x), the DOMAIN x is: -1 < x < 1. If the argument is outside this domain, ACOS returns zero and generates a "?DOMAIN error". The RANGE y is limited to the principal values of the Arc Cosine of x (ie. 0 =< y =< Pi). ;HelpStart IDENTITY,IDENTITIES Common mathematical identities with ACOS: ACOS(x) = Pi/2-ASIN(x) ACOS(-x) = Pi-ACOS(x) ACOS(1/x) = ASEC(x) ;HelpEnd IDENTITY,IDENTITIES ;HelpEnd ACOS,ACOS(X),ACOS2,ACOS2(X),ACOS3,ACOS3(X) ;HelpEnd FUNCTION,FUNCTIONS,NATIVE,TRIG,TRIGONOMETRIC ;********************************************************************** ;******* End of CALHELP2.CAL ******* ;**********************************************************************