The law of cosines states that in X Y Z x^2=y^2+z^2-2 y z cosX a. Explain how the law of cosines

step 1 So we can use the law of cosines to see if an angle is obtuse, acute, or right. So think about t

The law of cosines states that in X Y Z x^2=y^2+z^2-2 y z...

The law of cosines states that in $\triangle X Y Z$ $x^{2}=y^{2}+z^{2}-2 y z \cos X$ a. Explain how the law of cosines allows you to make a quick test to see if angle $X$ is acute, right, or obtuse, as shown: Property: Test for the Size of an Angle in a Triangle In $\triangle X Y Z:$ If $x^{2}<y^{2}+z^{2},$ then $x$ is an acute angle. If $x^{2}=y^{2}+z^{2},$ then $x$ is a right angle. If $x^{2}>y^{2}+z^{2},$ then $x$ is an obtuse angle. b. Without using your calculator, find whether angle $x$ is acute, right, or obtuse if $x=7 \mathrm{cm}$ $y=5 \mathrm{cm},$ and $z=4 \mathrm{cm}$.

The law of
cosines states that in $\triangle X Y Z$
$x^{2}=y^{2}+z^{2}-2 y z \cos X$
a. Explain how the law of cosines allows you to make a quick test to see if angle $X$ is acute, right, or obtuse, as shown:
Property: Test for the Size of an Angle in a Triangle
In $\triangle X Y Z:$
If $x^{2}<y^{2}+z^{2},$ then $x$ is an acute angle.
If $x^{2}=y^{2}+z^{2},$ then $x$ is a right angle.
If $x^{2}>y^{2}+z^{2},$ then $x$ is an obtuse angle.
b. Without using your calculator, find whether angle $x$ is acute, right, or obtuse if $x=7 \mathrm{cm}$ $y=5 \mathrm{cm},$ and $z=4 \mathrm{cm}$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental result in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem by accounting for non-right triangles, and expresses one side squared as the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle. This allows us to solve triangles when we have various combinations of sides and angles.

Testing Angle Type Using Side Lengths

By manipulating the Law of Cosines, one can compare the square of one side of a triangle with the sum of the squares of the other two sides to determine the nature of the corresponding angle. Specifically, if the square of the side is less than, equal to, or greater than the sum of the squares of the other sides, then the cosine of the angle is positive, zero, or negative, indicating that the angle is acute, right, or obtuse, respectively. This method provides a quick test to classify the angle without directly calculating the cosine value.

Transcript

Please give Ace some feedback