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Walchand College of Engineering (WCE), Sangli
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## Angle made by direction cosines of two lines in sine form Solution

STEP 0: Pre-Calculation Summary
Formula Used
angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2))
α = asin(sqrt(((l1*m2)- (l2*m1))^2+((m1*n2)-(m2*n1))^2+((n1*l2)-(n2*l1))^2))
This formula uses 3 Functions, 6 Variables
Functions Used
sin - Trigonometric sine function, sin(Angle)
asin - Inverse trigonometric sine function, asin(Number)
sqrt - Squre root function, sqrt(Number)
Variables Used
Direction cosine 1 with respect to x axis - Direction cosine 1 with respect to x axis is the cosine of angle made by a line w.r.to x axis. (Measured in Hundred)
Direction cosine 2 with respect to y axis - Direction cosine 2 with respect to y axis is the cosine of angle made by a line w.r.to y axis. (Measured in Hundred)
Direction cosine 2 with respect to x axis - Direction cosine 2 with respect to x axis is the cosine of angle made by a line w.r.to x axis. (Measured in Hundred)
Direction cosine 1 with respect to y axis - Direction cosine 1 with respect to y axis is the cosine of angle made by a line w.r.to y axis. (Measured in Hundred)
Direction cosine 2 with respect to z axis - Direction cosine 2 with respect to z axis is the cosine of angle made by a line w.r.to z axis. (Measured in Hundred)
Direction cosine 1 with respect to z axis - Direction cosine 1 with respect to z axis is the cosine of angle made by a line w.r.to z axis. (Measured in Hundred)
STEP 1: Convert Input(s) to Base Unit
Direction cosine 1 with respect to x axis: 0.7 Hundred --> 0.7 Hundred No Conversion Required
Direction cosine 2 with respect to y axis: 0.6 Hundred --> 0.6 Hundred No Conversion Required
Direction cosine 2 with respect to x axis: 0.7 Hundred --> 0.7 Hundred No Conversion Required
Direction cosine 1 with respect to y axis: 0.8 Hundred --> 0.8 Hundred No Conversion Required
Direction cosine 2 with respect to z axis: 0.6 Hundred --> 0.6 Hundred No Conversion Required
Direction cosine 1 with respect to z axis: 0.6 Hundred --> 0.6 Hundred No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
α = asin(sqrt(((l1*m2)- (l2*m1))^2+((m1*n2)-(m2*n1))^2+((n1*l2)-(n2*l1))^2)) --> asin(sqrt(((0.7*0.6)- (0.7*0.8))^2+((0.8*0.6)-(0.6*0.6))^2+((0.6*0.7)-(0.6*0.7))^2))
Evaluating ... ...
α = 0.185452088760136
STEP 3: Convert Result to Output's Unit
0.185452088760136 Radian -->10.6256219878433 Degree (Check conversion here)
10.6256219878433 Degree <-- Angle
(Calculation completed in 00.000 seconds)

## < 4 Angle in 3D space Calculators

Angle between two planes
angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))) Go
Angle made by direction cosines of two lines in sine form
angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2)) Go
Angle between two lines given direction ratios of two lines
angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2))) Go
Angle between line and plane given coefficients of line and plane
angle_a = asin(((Direction ratio 1 of line1*Direction Ratio 1)+(Direction ratio 2 of line1*Direction Ratio 2)+(Direction ratio 3 of line1*Direction Ratio 3))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction Ratio 1)^2+(Direction Ratio 2)^2+(Direction Ratio 3)^2))) Go

### Angle made by direction cosines of two lines in sine form Formula

angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2))
α = asin(sqrt(((l1*m2)- (l2*m1))^2+((m1*n2)-(m2*n1))^2+((n1*l2)-(n2*l1))^2))

## What is coordinate system in 3D space?

The three mutually perpendicular lines in a space which divides the space into eight parts and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system.

## How to Calculate Angle made by direction cosines of two lines in sine form?

Angle made by direction cosines of two lines in sine form calculator uses angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2)) to calculate the Angle, Angle made by direction cosines of two lines in sine form is defined as the sine value of angle made by two lines given their direction cosines. Angle and is denoted by α symbol.

How to calculate Angle made by direction cosines of two lines in sine form using this online calculator? To use this online calculator for Angle made by direction cosines of two lines in sine form, enter Direction cosine 1 with respect to x axis (l1), Direction cosine 2 with respect to y axis (m2), Direction cosine 2 with respect to x axis (l2), Direction cosine 1 with respect to y axis (m1), Direction cosine 2 with respect to z axis (n2) & Direction cosine 1 with respect to z axis (n1) and hit the calculate button. Here is how the Angle made by direction cosines of two lines in sine form calculation can be explained with given input values -> 10.62562 = asin(sqrt(((0.7*0.6)- (0.7*0.8))^2+((0.8*0.6)-(0.6*0.6))^2+((0.6*0.7)-(0.6*0.7))^2)).

### FAQ

What is Angle made by direction cosines of two lines in sine form?
Angle made by direction cosines of two lines in sine form is defined as the sine value of angle made by two lines given their direction cosines and is represented as α = asin(sqrt(((l1*m2)- (l2*m1))^2+((m1*n2)-(m2*n1))^2+((n1*l2)-(n2*l1))^2)) or angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2)). Direction cosine 1 with respect to x axis is the cosine of angle made by a line w.r.to x axis, Direction cosine 2 with respect to y axis is the cosine of angle made by a line w.r.to y axis, Direction cosine 2 with respect to x axis is the cosine of angle made by a line w.r.to x axis, Direction cosine 1 with respect to y axis is the cosine of angle made by a line w.r.to y axis, Direction cosine 2 with respect to z axis is the cosine of angle made by a line w.r.to z axis & Direction cosine 1 with respect to z axis is the cosine of angle made by a line w.r.to z axis.
How to calculate Angle made by direction cosines of two lines in sine form?
Angle made by direction cosines of two lines in sine form is defined as the sine value of angle made by two lines given their direction cosines is calculated using angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2)). To calculate Angle made by direction cosines of two lines in sine form, you need Direction cosine 1 with respect to x axis (l1), Direction cosine 2 with respect to y axis (m2), Direction cosine 2 with respect to x axis (l2), Direction cosine 1 with respect to y axis (m1), Direction cosine 2 with respect to z axis (n2) & Direction cosine 1 with respect to z axis (n1). With our tool, you need to enter the respective value for Direction cosine 1 with respect to x axis, Direction cosine 2 with respect to y axis, Direction cosine 2 with respect to x axis, Direction cosine 1 with respect to y axis, Direction cosine 2 with respect to z axis & Direction cosine 1 with respect to z axis and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Angle?
In this formula, Angle uses Direction cosine 1 with respect to x axis, Direction cosine 2 with respect to y axis, Direction cosine 2 with respect to x axis, Direction cosine 1 with respect to y axis, Direction cosine 2 with respect to z axis & Direction cosine 1 with respect to z axis. We can use 4 other way(s) to calculate the same, which is/are as follows -
• angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2))
• angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2)))
• angle_a = asin(((Direction ratio 1 of line1*Direction Ratio 1)+(Direction ratio 2 of line1*Direction Ratio 2)+(Direction ratio 3 of line1*Direction Ratio 3))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction Ratio 1)^2+(Direction Ratio 2)^2+(Direction Ratio 3)^2)))
• angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))) Let Others Know