Question Video: Using Direction Cosines to Evaluate Trigonometric Expressions Mathematics

Given that 𝑙, 𝑚, and 𝑛 are the direction cosines of a straight line, find the value of 𝑙² + 𝑚² + 𝑛².

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Video Transcript

Given that 𝑙, 𝑚, and 𝑛 are the direction cosines of a straight line, find the value of 𝑙 squared plus 𝑚 squared plus 𝑛 squared.

Let’s just consider that we have the space 𝑥, 𝑦, and 𝑧. Point 𝐴 is found at lowercase 𝑥𝑦𝑧. And we’ll have a line from the origin to point 𝐴, like this. Line 𝑂𝐴 is made up of three different vectors, here 𝑥 in pink, 𝑦 in yellow, and 𝑧 in green.

Under these conditions, we can let 𝑙 be equal to the cos of 𝛼, where 𝛼 is the angle measure between this pink line and our vector 𝑂𝐴. 𝑚 is gonna be equal to cos of 𝛽, where 𝛽 is the angle measure between the yellow line and blue line here. 𝑛 will be equal to the cosine of 𝛾, where 𝛾 is the angle between the green line and the blue line in our figure.

And the cosine value will be equal to the adjacent side length 𝑥 over the length of our line, which we find by saying 𝑥 squared plus 𝑦 squared plus 𝑧 squared and then taking the square root of that value. To find the cos of 𝛽, we’ll follow the same procedure. But this time, we’ll have the length 𝑦 as the numerator. And the cos of 𝛾 will be equal to 𝑧 over the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared.

Remember, we’re looking for 𝑙 squared plus 𝑚 squared plus 𝑛 squared. If we plug in the values that we know for 𝑙, 𝑚, and 𝑛, we need to take 𝑥 squared. And then, we need to square the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared. And that will give us 𝑥 squared over 𝑥 squared plus 𝑦 squared plus 𝑧 squared.

When we square 𝑚, we get 𝑦 squared over 𝑥 squared plus 𝑦 squared plus 𝑧 squared. And when we square 𝑛, we get 𝑧 squared over 𝑥 squared plus 𝑦 squared plus 𝑧 squared. All three of these fractions have a common denominator, which means we can add their numerators. 𝑥 squared plus 𝑦 squared plus 𝑧 squared can’t be simplified. So we just leave it 𝑥 squared plus 𝑦 squared plus 𝑧 squared. And the denominator doesn’t change. It’s also 𝑥 squared plus 𝑦 squared plus 𝑧 squared. And when the numerator equals the denominator, the value of that fraction is one.

We’ve just shown that 𝑙 squared plus 𝑚 squared plus 𝑛 squared equals one. In fact, the squares of the direction cosines of a straight line when added together always equal one.