Question

right triangles

1. evaluate to 4 decimal places: cos 12° 2
2. solve for each unknown value to one decimal place.6

a) cos 12° = \\(\frac{z}{5}\\) b) tan 38° = \\(\frac{15}{x}\\) c) sin \\(\theta\\) = 0.7531

Explanation:

Step1: Recall trigonometric - inverse function relationship

For $\cos12^{\circ}$, we know that $\cos12^{\circ}\approx0.9781$.

Step2: Solve for $x$ in $\cos12^{\circ}=\frac{5}{x}$

Cross - multiply to get $x\cos12^{\circ}=5$, then $x = \frac{5}{\cos12^{\circ}}$. Substitute $\cos12^{\circ}\approx0.9781$, so $x=\frac{5}{0.9781}\approx5.1$.

Step3: Solve for $x$ in $\tan38^{\circ}=\frac{15}{x}$

Cross - multiply to get $x\tan38^{\circ}=15$, then $x=\frac{15}{\tan38^{\circ}}$. Since $\tan38^{\circ}\approx0.7813$, $x=\frac{15}{0.7813}\approx19.2$.

Step4: Solve for $\theta$ in $\sin\theta = 0.7531$

Use the inverse sine function, $\theta=\sin^{-1}(0.7531)$. So $\theta\approx48.8^{\circ}$.

Answer:

1. $\cos12^{\circ}\approx0.9781$
2. a) $x\approx5.1$

b) $x\approx19.2$
c) $\theta\approx48.8^{\circ}$