The Law of Cosines
The law of cosines relates all three sides of a triangle to one angle, and it works where the law of sines cannot: when you know three sides, or two sides and the angle between them. With a right angle it reduces to the Pythagorean theorem.
The Law of Cosines, Example 2:
(b=20, a=8, c=14)
Tutor-style math help
The Law of Cosines: what to notice and how to work it
Trigonometry skill
Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.
What to notice first
Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.
Common student mistake
Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.
Key formulas and cues
(sintheta=frac{text{opposite}}{text{hypotenuse}})
(costheta=frac{text{adjacent}}{text{hypotenuse}})
(tantheta=frac{sintheta}{costheta})
(sin^2theta+cos^2theta=1)
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
Example: opposite = 5, hypotenuse = 13
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Answer: (sintheta=frac5{13})
Unit-circle cosine
Example: (cos(0))
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Answer: (1)
Try one before moving on
Try: In a right triangle, tangent equals which ratio?
Answer: Opposite over adjacent.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
The Law of Cosines: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Cosine in a right triangle is:
2. (180^circ) equals:
3. The unit circle has radius:
(cos B= frac {14^2+8^2-20^2}{2(14)(8)} =frac {196+ 64 to 400}{176}=frac{-140}{224}=-0.625)
Since (cosB) is negative, (B) is an obtuse angle.
(B≅128.69 ^circ )
Exercises for the Law of Cosines
In the ABC triangle, find the side of c.
1.
2.
3.
- (color{blue}{31.12})
- (color{blue}{44.68})
- (color{blue}{21.49})
Original price was: $109.99.$54.99Current price is: $54.99.
Original price was: $109.99.$54.99Current price is: $54.99.
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