Math · Precalculus · Worked example

Solve an SAS triangle with the law of cosines

A triangle has sides $a = 3$ and $b = 8$ , and the angle between them is $C = 60 °$ . Find side $c$ and angles $A$ and $B$ .

Choose the law

Two sides and the angle between them is SAS. No side is paired with its opposite angle yet, so the law of sines cannot start. The law of cosines gives the side opposite the known angle.

c^{2} = a^{2} + b^{2} - 2 a b cos C

Find side $c$

Substitute $a = 3$ , $b = 8$ and $C = 60 °$ . Since $cos 60 ° = 1 / 2$ , the last term is $2 \cdot 3 \cdot 8 \cdot (1 / 2) = 24$ .

\begin{array}{l} c^{2} = 3^{2} + 8^{2} - 2 \cdot 3 \cdot 8 cos 60^{\circ} \\ c^{2} = 9 + 64 - 24 \\ c^{2} = 49 \\ c = 7 \end{array}

Find $sin A$ with the law of sines

Side $c$ and angle $C$ are now a complete pair. Find angle $A$ first: it faces the shortest side, so it is acute.

\begin{array}{l} \frac{sin A}{3} = \frac{sin 60^{\circ}}{7} \\ sin A = \frac{3 sin 60^{\circ}}{7} \\ sin A \approx 0.3712 \end{array}

Take the inverse sine

$A = {sin}^{- 1} (0.3712)$ $\approx 21.8 °$ . To check it, take the sine of the answer: $sin 21.8 ° \approx 0.3714$ , which agrees with $0.3712$ to three figures.

A \approx {21.8}^{\circ}

Finish with the angle sum

The angles add to $180 °$ . $B$ turns out obtuse, which an inverse sine could not have reported. That is why the smaller angle came first.

\begin{array}{l} B = 180^{\circ} - 60^{\circ} - {21.8}^{\circ} \\ B = {98.2}^{\circ} \end{array}

Result

$c = 7$ , $A \approx 21.8 °$ and $B \approx 98.2 °$ .

Your turn

1. One like the example

Two sides of a triangle, $5$ and $8$ , meet at a $60 °$ angle. Find the third side.

Show the answer and explanation

$c = 7$ .

Two sides and the angle between them is SAS, so $c^{2} = 5^{2} + 8^{2}$ $- 2 \cdot 5 \cdot 8 \cdot cos 60 °$ $= 25 + 64 - 40$ $= 49$ , and $c = 7$ . The cosine of $60 °$ is $1 / 2$ , which is why the last term is $40$ .

\begin{array}{l} c^{2} = 5^{2} + 8^{2} - 2 \cdot 5 \cdot 8 cos 60^{\circ} \\ c^{2} = 25 + 64 - 40 \\ c^{2} = 49 \\ c = 7 \end{array}

2. A variation

A triangle has sides $5$ , $7$ and $8$ . Find the angle opposite the side of length $7$ .

Show the answer and explanation

$60 °$ .

Three sides is SSS, so use the law of cosines solved for the cosine, $cos C$ $= (5^{2} + 8^{2} - 7^{2}) / (2 \cdot 5 \cdot 8)$ $= 40 / 80$ $= 1 / 2$ . The angle between $0 °$ and $180 °$ with cosine $1 / 2$ is $60 °$ .

\begin{array}{l} cos C = \frac{5^{2} + 8^{2} - 7^{2}}{2 \cdot 5 \cdot 8} \\ cos C = \frac{40}{80} \\ cos C = \frac{1}{2} \end{array}

3. Somewhere new

A baseball diamond is a square $90 ft$ on a side. The pitcher’s mound is $60.5 ft$ from home plate, on the line from home to second base, and that line makes a $45 °$ angle with the line from home to first. How far is the mound from first base?

Show the answer and explanation

About $63.7 ft$ .

Home plate, the mound and first base form a triangle with sides $90 ft$ and $60.5 ft$ around a $45 °$ angle, an SAS case. $d^{2} = 90^{2} + {60.5}^{2}$ $- 2 (90) (60.5) cos 45 °$ $\approx 11, 760.25 - 7700.39$ $\approx 4059.9$ , so $d \approx 63.7 ft$ .

\begin{array}{l} d^{2} = 90^{2} + {60.5}^{2} - 2 \cdot 90 \cdot 60.5 cos 45^{\circ} \\ d^{2} \approx 4059.9 \\ d \approx 63.7 \end{array}

\begin{array}{l} d^{2} = 90^{2} + {60.5}^{2} \\ - 2 \cdot 90 \cdot 60.5 cos 45^{\circ} \\ d^{2} \approx 4059.9 \\ d \approx 63.7 \end{array}

Keep exploring

Open the side in Math and delete the degree sign from $cos 60 °$ . The next row turns to ✗, and Chalk says the $60$ in row 1 is read in radians. Put the sign back, then change $24$ to $48$ , which is what forgetting $cos 60 °$ gives. That row reads ✗ too.

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Practice the key terms as flashcards, then open the example in its tool, change a value and keep a useful result on your board.

Study the 5 key terms Check side c in Math Check sin A in Math Open worked example on a board Law of cosines in Math Reference

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Choose the law

Find side cc

Find sinAsinA with the law of sines

Take the inverse sine

Finish with the angle sum

Result

Your turn

1. One like the example

c=7c=7.

2. A variation

60⁢°60°.

3. Somewhere new

About 63.7 ft63.7 ft.

Keep exploring

Study it, then try it

Find side $c$

Find $sin A$ with the law of sines

$c = 7$ .

$60 °$ .

About $63.7 ft$ .