Download Law of Sines

Law of Sines Proof
Name:
_______________________
CAPITAL LETTERS are used for angles, and lower-case letters are used for side lengths.
ΔABC shown to the right has an altitude k that is perpendicular to the
base.
1. The altitude creates two right triangles inside ΔABC. Using right
triangle trigonometry, write two equations, one involving sin A,
and one involving sin C. (lower case letters)
( sin A ) =
k
( sin C ) =
2. Notice that each of the equations involves k. Solve each equation for k.
3. Since both equations are equal to k, they can be set equal to each other. This is which
property of equality?
4. Set the equations equal to each other to form a new equation that no longer involves k.
5. Divide both sides by ac (lower case) and simplify.
Now ΔABC has an altitude p drawn from a different angle.
6. Using right triangle trigonometry, write two equations, one
involving sin A and one involving sin B.
( sin A )=
p
( sin B )=
7. Solve for p in both equations.
8. Set both equations equal to each other, divide by ab (lower case), and simplify.
9. Use the equations in Question 5 and Question 8 to write one big relationship between all 3
sides and all 3 angles. Hooray, you proved the Law of Sines (see top of page).
Law of Cosines Proof
c2 = a2 + b2 – 2ab(cos C)
In ΔABC, altitude h is drawn from B and separates side
b into segments x and b – x.
1. Use the Pythagorean Theorem to write two
equations...
One involving a, x, and h.
And another involving c, h, and (b-x). Don’t expand (b-x)2 yet
2. Solve both equations for h2 (do not square root).
3. Apply the transitive property to eliminate h2.
4. Solve for c2.
5. Expand (b-x)2 to b2 - 2bx+x2 in the above equation and then simplify.
6. Compare your equation to the Law of Cosines at the top of the page. What variable still needs
to be eliminated from your equation? ____.
7. On the right triangle CDB, write a relationship involving x and cos C.
8. Solve for x.
9. Substitute this expression in for x in the equation from question 5. Hooray, you proved the
Law of Cosines.