3. Law of Cosines
Why do we need it? How is it derived from what we already know?
We need law of cosines to find the third side when knowing two sides and an angle (SSA) or knowing all three sides (SSS) to find the angles.
We got a triangle label ABC that's not a right angle. Angle A's point is (0,0), angle B's point is (ccosA,csinA), ccosA is the x value because cosA equals d/c and csinA is the y value because sinA equals h/c. Angle C's point is (b,0), b is the distance across from angle B between angle A and C. Draw a line down angle B to make h, h is the drawn height of the triangle. With that it makes a 90 degrees angle. Then label side c and d also ccosA. Side c is the distance across from angle C between angle A and B. Side d is the distance across from angle B in just the right triangle. Side a is the distance between angle B and C (see picture 1). Use the Pythagorean Theorem in triangle CBD but substitute h to csinA and r to ccosA. Foil to get b^2-2bccosA+c^2cos^2A (see picture 2). Put cos^2A + sin^2A in parenthesis, since since cos^2A + sin^2A equals one. Finally get the formula (see picture 3), it can also be used to produce other letters statements (see picture 4). That's how we derive law of cosines.
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| Picture 1 |
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| Picture 2 |
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| Picture 3 |
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| Picture 4 |
http://www.regentsprep.org/Regents/math/algtrig/ATT12/derivelawofsines.htm
5. Area formulas
Draw out a right triangle with angle A equal to 35 and angle B equal to 65. Side b equal to 4. Use the law of sines to get side b and c. Use the area of an oblique triangle, and Heron’s area formula to get both areas (see picture 5).
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| Picture 5 |
Work cited
http://www.regentsprep.org/Regents/math/algtrig/ATT12/derivelawofsines.htm
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