How To Find An Angle Using Law Of Cosines

Law of Cosines

The Law of Cosines is used to notice the remaining parts of an oblique (non-right) triangle when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known.  In either of these cases, information technology is impossible to use the Law of Sines because we cannot prepare a solvable proportion.

The Constabulary of Cosines states:

$c^{\mathrm{two}}=a^{\mathrm{two}}+b^2-2ab\cos C$ .

This resembles the Pythagorean Theorem except for the 3rd term and if $C$ is a right angle the 3rd term equals $0$ because the cosine of $90°$ is $0$ and we get the Pythagorean Theorem.  So, the Pythagorean Theorem is a special case of the Police of Cosines.

The Constabulary of Cosines tin likewise be stated as

$b^2=a^2+c^2-\mathrm{ii}ac\cos B$ or

$a^2=b^2+c^2-2bc\cos A$ .

Example 1: 2 Sides and the Included Angle-SAS

Given $a=11,b=5$ and $\mathrm{1000}\angle C=20°$ . Find the remaining side and angles.

$c^{\mathrm{ii}}=a^2+b^2-2ab\cos C$

$c=\sqrt{a^2+b^2-2ab\cos C}$

$=\sqrt{{11}^2+5^{\mathrm{ii}}-\mathrm{ii}\left(11\right)\left(5\right)\left(\cos 20°\right)}$

$\approx \mathrm{six.53}$

To find the remaining angles, information technology is easiest to now employ the Constabulary of Sines.

$\sin A\approx \frac{\mathrm{xi}\sin \mathrm{twenty}°}{6.53}$

$A\approx 144.82°$

$\sin B\approx \frac{5\sin \mathrm{twenty}°}{6.53}$

$B\approx \mathrm{xv.ii}°$

Note that angle $A$ is opposite to the longest side and the triangle is non a right triangle. So, when you have the inverse you need to consider the obtuse bending whose sine is $\frac{11\sin \left(\mathrm{twenty}°\right)}{6.53}\approx 0.5761$ .

Example two: Three Sides-SSS

Given $a=8,b=19$ and $c=\mathrm{xiv}$ .  Discover the measures of the angles.

It is all-time to observe the angle opposite the longest side first.  In this example, that is side $b$ .

$\cos B=\frac{b^2-a^2-c^2}{-2ac}=\frac{{19}^2-{\mathrm{eight}}^2-{\mathrm{fourteen}}^2}{-2\left(8\right)\left(14\right)}\approx -0.45089$

Since $\cos B$ is negative, we know that $B$ is an obtuse angle.

$B\approx 116.80°$

Since $B$ is an obtuse angle and a triangle has at well-nigh 1 birdbrained bending, we know that bending $A$ and angle $C$ are both acute.

To discover the other two angles, information technology is simplest to utilize the Law of Sines.

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

$\frac{8}{\sin A}\approx \frac{19}{\sin 116.80°}\approx \frac{\mathrm{fourteen}}{\sin C}$

$\sin A\approx \frac{\mathrm{viii}\sin 116.80°}{19}$

$A\approx 22.08°$

$\sin C\approx \frac{\mathrm{xiv}\sin 116.80°}{\mathrm{nineteen}}$

$C\approx 41.12°$

How To Find An Angle Using Law Of Cosines,

Source: https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-cosines

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