Determine the unknown side lengths

Question

Rounded to nearest tenth

solving for $y$:

tan = opposite / adjacent

12tan(53°) = 15.9 cm

solving for $z$:

cos = adjacent / hypotenuse

12cos(53°) = 9.6 cm

Am I correct? Also, the thing that confuses me is the fact that I got $15.9$ for $y$ when clearly the side that reads $12$ cm is bigger; for $z$ I got $9.6$ but the hypotenuse is bigger than sides $z$ and $y$. Any explanation to help settle my confusion would help a lot.

Answer 1

You need to be careful which lengths you plug in to the equation. For the $53^{\circ}$ angle, the "adjacent" side is $y$, the "opposite" side is $12 \ \text{cm}$ and the hypotenuse is $z$. Therefore, you should get:

$\tan 53^{\circ} = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{12 \ \text{cm}}{y} \leadsto y = \dfrac{12 \ \text{cm}}{\tan 53^{\circ}}$

and

$\cos 53^{\circ} = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{y}{z} \leadsto z = \dfrac{y}{\cos 53^{\circ}}$

You can also use $\sin 53^{\circ} = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{12 \ \text{cm}}{z} \leadsto z = \dfrac{12 \ \text{cm}}{\sin 53^{\circ}}$.

Answer 2

You can apply sine rule $$\frac{\sin{(A)}}{a}=\frac{\sin{(B)}}{b}=\frac{\sin{(C)}}{c}$$ and get the answer easy to solve.

Answer 3

Hint: $tan(53)=\frac{4}{3},cos(53)=\frac{3}{5}$ thats all waht you need to solve it. no need to apply cosine,sine rules here.