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If a line makes angles α,β,γ with the +ve direction of x,y and z axis respectively, then cos2α+cos2β+cos2γ
- a)1/2
- b)-1/2
- c)-1
- d)1
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If a line makes angles α,β,γ with the +ve direction o...
Of $\alpha$ and $\beta$ with the positive $x$ and $y$ axes, respectively, then we can use trigonometry to find the slope of the line. We have:
$\tan \alpha = \frac{\text{slope}}{1} = \text{slope}$
and
$\tan \beta = \frac{1}{\text{slope}}$
Solving for $\text{slope}$ in the second equation gives:
$\text{slope} = \frac{1}{\tan \beta}$
Substituting this into the first equation, we get:
$\tan \alpha = \frac{1}{\tan \beta}$
Solving for $\tan \alpha$, we have:
$\tan \alpha = \cot \beta$
Finally, we can use the identity $\cot \theta = \frac{1}{\tan \theta}$ to get:
$\tan \alpha = \frac{1}{\tan \beta} = \frac{1}{\cot \alpha} = \tan \left(\frac{\pi}{2} - \alpha\right)$
Therefore, the slope of the line is equal to the tangent of the angle it makes with the positive $x$ axis, which is also equal to the tangent of the complementary angle to the angle it makes with the positive $y$ axis.
$\tan \alpha = \frac{\text{slope}}{1} = \text{slope}$
and
$\tan \beta = \frac{1}{\text{slope}}$
Solving for $\text{slope}$ in the second equation gives:
$\text{slope} = \frac{1}{\tan \beta}$
Substituting this into the first equation, we get:
$\tan \alpha = \frac{1}{\tan \beta}$
Solving for $\tan \alpha$, we have:
$\tan \alpha = \cot \beta$
Finally, we can use the identity $\cot \theta = \frac{1}{\tan \theta}$ to get:
$\tan \alpha = \frac{1}{\tan \beta} = \frac{1}{\cot \alpha} = \tan \left(\frac{\pi}{2} - \alpha\right)$
Therefore, the slope of the line is equal to the tangent of the angle it makes with the positive $x$ axis, which is also equal to the tangent of the complementary angle to the angle it makes with the positive $y$ axis.
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Community Answer
If a line makes angles α,β,γ with the +ve direction o...
Of 30° and 60° with the x-axis, then its slope can be found using trigonometry.
Let's call the line AB, where A is the point where the line intersects the x-axis and B is the point where it intersects the y-axis.
The angle between the line and the x-axis is 30°, which means that the angle between the line and the y-axis is 90° - 30° = 60°.
Using trigonometry, we can find the slope of the line as follows:
tan(30°) = AB/BA
tan(60°) = BA/AB
We can rearrange these equations to solve for AB/BA and BA/AB:
AB/BA = tan(30°)
BA/AB = tan(60°)
Since slope is defined as the change in y divided by the change in x, we can use these ratios to find the slope of the line:
slope = (change in y) / (change in x)
From the coordinates of points A and B, we know that the change in x is BA and the change in y is AB.
Therefore, the slope of the line is:
slope = AB/BA = tan(30°) = 1/sqrt(3)
or
slope = BA/AB = tan(60°) = sqrt(3)
So the line has two possible slopes depending on how you define it, either 1/sqrt(3) or sqrt(3).
Let's call the line AB, where A is the point where the line intersects the x-axis and B is the point where it intersects the y-axis.
The angle between the line and the x-axis is 30°, which means that the angle between the line and the y-axis is 90° - 30° = 60°.
Using trigonometry, we can find the slope of the line as follows:
tan(30°) = AB/BA
tan(60°) = BA/AB
We can rearrange these equations to solve for AB/BA and BA/AB:
AB/BA = tan(30°)
BA/AB = tan(60°)
Since slope is defined as the change in y divided by the change in x, we can use these ratios to find the slope of the line:
slope = (change in y) / (change in x)
From the coordinates of points A and B, we know that the change in x is BA and the change in y is AB.
Therefore, the slope of the line is:
slope = AB/BA = tan(30°) = 1/sqrt(3)
or
slope = BA/AB = tan(60°) = sqrt(3)
So the line has two possible slopes depending on how you define it, either 1/sqrt(3) or sqrt(3).
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If a line makes angles α,β,γ with the +ve direction of x,y and z axis respectively, then cos2α+cos2β+cos2γa)1/2b)-1/2c)-1d)1Correct answer is option 'D'. Can you explain this answer?
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