LEMMA

Advanced trigonometry

Laws of sines and cosines

ALEKS placement

Use the law of sines when a side pairs with its opposite angle; otherwise use the law of cosines.

What this covers

sinAa​=sinBb​,c2=a2+b2−2abcosC
  • Count what you know: two angles and a side, or two sides and an angle.
  • A complete angle-side pair points to the law of sines.
  • Two sides plus the included angle, or all three sides, point to the law of cosines.
  • For two sides and a non-included angle, check the inverse-sine angle and its supplement; each must leave a positive third angle.
  • A sine ratio above 1 means no triangle fits. If the two angle candidates coincide, count only one.

Worked example

Worked example

Find c when a = 7, b = 9, and C = 40°.

  1. c2=49+81−2(7)(9)cos40∘Two sides and the included angle means the law of cosines.
  2. c≈5.8Evaluate and take the square root.
Another worked example

In a triangle, A = 30°, B = 45°, and a = 6. Find b.

  1. sin45∘b​=sin30∘6​Each side must pair with the sine of its opposite angle.
  2. b=61/2√2/2​=6√2Multiply by sin 45° and substitute both exact special-angle values.

A common mistake

A mistake Lemma catches

From

c2=(130−126)cos40∘c2=130−126cos40∘

“The cosine multiplies only 2ab = 2 · 7 · 9 = 126, so 126 cos 40° is subtracted as a whole: c² = 130 − 126 cos 40° ≈ 130 − 96.5 = 33.5, and c ≈ 5.8. Subtracting first, (130 − 126) cos 40° = 4 cos 40° ≈ 3.06, gives c ≈ 1.7 — too short to close a triangle with sides 7 and 9.”

Try one

Sample problem

A rectangular field has perpendicular sides 3 m and 4 m. Find its diagonal c.

c2=32+42−2(3)(4)cos(90∘)

Practice oblique triangles free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this