A triangle has sides A, B, and C. Sides A and B are of lengths $2$ $2$ and $5$ $5$ , respectively, and the angle between A and B is $\frac{π}{12}$ $pi/12$ . What is the length of side C?

Question

1553 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-01-27T10:22:01

$C \approx 4.4$ $C ~~4.4$

Sketch Sketch
Use the trigonometric relationships e.g. sine = opposite /hypotenuse
$A = x + y$ $A = x + y$

$\frac{x}{B} = cos (\frac{π}{12})$ $x/B = cos(pi/12)$

$:. x= Bcos(pi/12) = 5cos(pi/12)$

Similarly $h = 5sin(pi/12)$

$y = A - x = 2 - 5cos(pi/12)$

Using Pythagoras, $C^2 = h^2 + y^2$

:. $C^2 = (5sin(pi/12))^2 +(2-5cos(pi/12))^2$

$C^2 = 25sin^2(pi/12) +4 - 10cos(pi/12) +25cos^2(pi/12)$

$C^2 = 25(sin^2(pi/12) +cos^2(pi/12)) +4 -10cos(pi/12)$

Using the fact that $sin^2(theta) + cos^2(theta) = 1$ this is now

$C^2 = 25 + 4 - 10cos(pi/12) =29 - 10cos(pi/12)$

$C~~ sqrt(29-9.659) ~~4.4$

A triangle has sides A, B, and C. Sides A and B are of lengths 222 and 555, respectively, and the angle between A and B is pi/12π12pi/12. What is the length of side C?