Week 17 – Precalc 11

One of the things we learned this week in Precalc 11 is how to use the cosine law to solve for angles or sidelengths of a triangle.

Cosine Law Formula

Sidelength Angle
a^2=b^2+c^2-2bc CosA Cos A=\frac{b^2+c^2-a^2}{2bc}
b^2=a^2+c^2-2ac Cos B Cos B=\frac{a^2+c^2-b^2}{2ac}
c^2=a^2+b^2-2ab Cos C Cos C=\frac{a^2+b^2-c^2}{2ab}


Since there are two laws, Sine and Cosine. To determine which law to use, look at one of the angles if there is one given, if the opposite side has a given sidelength then use the Sine Law, if the opposite side has an unknown sidelength then use the Cosine Law.

Solving for an angle

Solve for ∠B.
First we need to figure out what formula to use.
Next, we can fill in the the values.
Then we can do the calculations.

Solving for a sidelength

Solve for side c.
First figure out what formula to use.
Next, fill in the values.
Then do the calculations.


In △ABC, AB=7, ∠B=105°, and BC=10; determine side b to one decimal place.

first we need to draw what the triangle might look like, it doesn’t have to be accurate.

b^2=a^2+c^2-2ac Cos B
b^2=10^2+7^2-2(10)(7)Cos 105°

Solve for ∠Z to the nearest degree.

Cos Z = \frac{x^2+y^2-z^2}{2xy}
Cos Z = \frac{23^2+11^2-14.7^2}{2(23)(11)}
Cos Z = \frac{433.91}{506}
Cos Z = 0.857
Z = Cos^{-1}(0.857)
Z = 31°

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