# Question Video: Using Direction Cosines to Evaluate Trigonometric Expressions Mathematics

Given that π, π, and π are the direction cosines of a straight line, find the value of πΒ² + πΒ² + πΒ².

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### Video Transcript

Given that π, π, and π are the direction cosines of a straight line, find the value of π squared plus π squared plus π squared.

Letβs just consider that we have the space π₯, π¦, and π§. Point π΄ is found at lowercase π₯π¦π§. And weβll have a line from the origin to point π΄, like this. Line ππ΄ is made up of three different vectors, here π₯ in pink, π¦ in yellow, and π§ in green.

Under these conditions, we can let π be equal to the cos of πΌ, where πΌ is the angle measure between this pink line and our vector ππ΄. π is gonna be equal to cos of π½, where π½ is the angle measure between the yellow line and blue line here. π will be equal to the cosine of πΎ, where πΎ 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 π₯ over the length of our line, which we find by saying π₯ squared plus π¦ squared plus π§ squared and then taking the square root of that value. To find the cos of π½, weβll follow the same procedure. But this time, weβll have the length π¦ as the numerator. And the cos of πΎ will be equal to π§ over the square root of π₯ squared plus π¦ squared plus π§ squared.

Remember, weβre looking for π squared plus π squared plus π squared. If we plug in the values that we know for π, π, and π, we need to take π₯ squared. And then, we need to square the square root of π₯ squared plus π¦ squared plus π§ squared. And that will give us π₯ squared over π₯ squared plus π¦ squared plus π§ squared.

When we square π, we get π¦ squared over π₯ squared plus π¦ squared plus π§ squared. And when we square π, we get π§ squared over π₯ squared plus π¦ squared plus π§ squared. All three of these fractions have a common denominator, which means we can add their numerators. π₯ squared plus π¦ squared plus π§ squared canβt be simplified. So we just leave it π₯ squared plus π¦ squared plus π§ squared. And the denominator doesnβt change. Itβs also π₯ squared plus π¦ squared plus π§ squared. And when the numerator equals the denominator, the value of that fraction is one.

Weβve just shown that π squared plus π squared plus π squared equals one. In fact, the squares of the direction cosines of a straight line when added together always equal one.