Question

1. given that $\theta = 35^\circ$ and $r = 8$ in the triangle above, find the value of $x$ and $y$. remember to put your calculator in degree mode. circle the correct answer below. (10 points)

triangle image
a) $x = 6.55, y = 5.60$ b) $x = 6.55, y = 4.59$
c) $x = 4.59, y = 6.55$ d) $x = 4.59, y = 5.60$

1. find the missing measures below. show your work and round answers to one decimal place if necessary. (10 points each blank)

right triangle image with leg 4, angle 34°
$x = \underline{\quad\quad\quad}$
$y = \underline{\quad\quad\quad}$
$z = \underline{\quad\quad\quad}$

1. use the law of cosines to solve for the indicated measure. show your work and round to one decimal place if necessary. (10 points)

triangle image with sides 10, 4, angle 50°
$x = \underline{\quad\quad\quad}$

Explanation:

Problem 1

Step1: Identify triangle type (right triangle)

The triangle is right - angled, so we can use trigonometric ratios. Let's assume \(r\) is the hypotenuse (\(r = 8\)), \(\theta=35^{\circ}\). For the right - triangle, \(\cos\theta=\frac{x}{r}\) and \(\sin\theta=\frac{y}{r}\) (where \(x\) is adjacent to \(\theta\) and \(y\) is opposite to \(\theta\)).

Step2: Calculate \(x\)

Using the cosine ratio: \(\cos\theta=\frac{x}{r}\), so \(x = r\times\cos\theta\). Substitute \(r = 8\) and \(\theta = 35^{\circ}\). \(\cos(35^{\circ})\approx0.8192\), then \(x=8\times0.8192 = 6.5536\approx6.55\).

Step3: Calculate \(y\)

Using the sine ratio: \(\sin\theta=\frac{y}{r}\), so \(y = r\times\sin\theta\). \(\sin(35^{\circ})\approx0.5736\), then \(y = 8\times0.5736=4.5888\approx4.59\).

The triangle is a right - triangle with one angle \(34^{\circ}\) and the opposite side to \(34^{\circ}\) is \(4\) (the vertical leg), the adjacent side is \(x\) (horizontal leg), the hypotenuse is \(z\), and the other non - right angle \(y=90 - 34=56^{\circ}\)

Step1: Find \(x\)

We know that \(\tan(34^{\circ})=\frac{4}{x}\) (since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)). So \(x=\frac{4}{\tan(34^{\circ})}\). \(\tan(34^{\circ})\approx0.6745\), then \(x=\frac{4}{0.6745}\approx5.9\)

Step2: Find \(y\)

\(y = 90^{\circ}-34^{\circ}=56^{\circ}\)

Step3: Find \(z\)

We know that \(\sin(34^{\circ})=\frac{4}{z}\) (since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)). So \(z=\frac{4}{\sin(34^{\circ})}\). \(\sin(34^{\circ})\approx0.5592\), then \(z=\frac{4}{0.5592}\approx7.2\)

The Law of Cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\), where \(a = 10\), \(b = 4\), and \(C = 50^{\circ}\), and we want to find \(x\) (the side opposite angle \(C\)).

Step1: Apply Law of Cosines

\(x^{2}=10^{2}+4^{2}-2\times10\times4\times\cos(50^{\circ})\)

Step2: Calculate each term

\(10^{2}=100\), \(4^{2} = 16\), \(2\times10\times4 = 80\), \(\cos(50^{\circ})\approx0.6428\)

Step3: Substitute values

\(x^{2}=100 + 16-80\times0.6428=116 - 51.424 = 64.576\)

Step4: Find \(x\)

\(x=\sqrt{64.576}\approx8.0\)

Answer:

b) \(x = 6.55,y = 4.59\)

Problem 2