Calculate the angles in the figure below.

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Here are the solutions to the problems:

1) Calculate the angles in the figure below.

a) Calculate $\alpha$

Identify the known sides relative to angle $\alpha$ .
The side opposite $\alpha$ is 3.
The side adjacent to $\alpha$ is 4.
Use the tangent function.
$\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}$ $\tan(\alpha) = \frac{3}{4}$
Calculate $\alpha$ .
$\alpha = \arctan\left(\frac{3}{4}\right)$ $\alpha \approx 36.87^\circ$
The angle is \boxed{\text{36.87^\circ}}.

b) Calculate $\beta$
Identify the known sides relative to angle $\beta$ .
The side opposite $\beta$ is 2.
The side adjacent to $\beta$ is 7.
Use the tangent function.
$\tan(\beta) = \frac{\text{opposite}}{\text{adjacent}}$ $\tan(\beta) = \frac{2}{7}$
Calculate $\beta$ .
$\beta = \arctan\left(\frac{2}{7}\right)$ $\beta \approx 15.95^\circ$
The angle is \boxed{\text{15.95^\circ}}.

c) Calculate $\phi$
Identify the known sides relative to angle $\phi$ .
The side opposite $\phi$ is 4.
The hypotenuse is 5.
Use the sine function.
$\sin(\phi) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin(\phi) = \frac{4}{5}$
Calculate $\phi$ .
$\phi = \arcsin\left(\frac{4}{5}\right)$ $\phi \approx 53.13^\circ$
The angle is \boxed{\text{53.13^\circ}}.

2) Find the lettered sides in the figure below.

a) Find $x$
Identify the known angle and sides relative to $x$ .
The angle is $30^\circ$ .
The side adjacent to $30^\circ$ is 4.
The hypotenuse is $x$ .
Use the cosine function.
$\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\cos(30^\circ) = \frac{4}{x}$
Solve for $x$ .
$x = \frac{4}{\cos(30^\circ)}$ $x = \frac{4}{\frac{\sqrt{3}}{2}}$ $x = \frac{8}{\sqrt{3}}$ $x = \frac{8\sqrt{3}}{3}$ $x \approx 4.62$
The side length is $\boxed{\text{4.62}}$ .

b) Find $y$
Identify the known angle and sides relative to $y$ .
The angle is $47^\circ$ .
The side opposite $47^\circ$ is $y$ .
The side adjacent to $47^\circ$ is 5.
Use the tangent function.
$\tan(47^\circ) = \frac{\text{opposite}}{\text{adjacent}}$ $\tan(47^\circ) = \frac{y}{5}$
Solve for $y$ .
$y = 5 \times \tan(47^\circ)$ $y \approx 5 \times 1.0724$ $y \approx 5.36$
The side length is $\boxed{\text{5.36}}$ .

c) Find $z$
Identify the known angle and sides relative to $z$ .
The angle is $52^\circ$ .
The side opposite $52^\circ$ is $z$ .
The hypotenuse is 7.
Use the sine function.
$\sin(52^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin(52^\circ) = \frac{z}{7}$
Solve for $z$ .
$z = 7 \times \sin(52^\circ)$ $z \approx 7 \times 0.7880$ $z \approx 5.52$
The side length is $\boxed{\text{5.52}}$ .

Calculate the angles in the figure below.

Step-by-step solution

Identify the known sides relative to angle $\alpha$ .

Use the tangent function.

Calculate $\alpha$ .

Identify the known sides relative to angle $\beta$ .

Use the tangent function.

Calculate $\beta$ .

Identify the known sides relative to angle $\phi$ .

Use the sine function.

Calculate $\phi$ .

Identify the known angle and sides relative to $x$ .

Use the cosine function.

Solve for $x$ .

Identify the known angle and sides relative to $y$ .

Use the tangent function.

Solve for $y$ .

Identify the known angle and sides relative to $z$ .

Use the sine function.

Solve for $z$ .

Still stuck on this one?

More Mathematics Questions

Calculate the angles in the figure below.

Step-by-step solution

Identify the known sides relative to angle α\alphaα.

Use the tangent function.

Calculate α\alphaα.

Identify the known sides relative to angle β\betaβ.

Use the tangent function.

Calculate β\betaβ.

Identify the known sides relative to angle ϕ\phiϕ.

Use the sine function.

Calculate ϕ\phiϕ.

Identify the known angle and sides relative to xxx.

Use the cosine function.

Solve for xxx.

Identify the known angle and sides relative to yyy.

Use the tangent function.

Solve for yyy.

Identify the known angle and sides relative to zzz.

Use the sine function.

Solve for zzz.

Still stuck on this one?

More Mathematics Questions

Identify the known sides relative to angle $\alpha$ .

Calculate $\alpha$ .

Identify the known sides relative to angle $\beta$ .

Calculate $\beta$ .

Identify the known sides relative to angle $\phi$ .

Calculate $\phi$ .

Identify the known angle and sides relative to $x$ .

Solve for $x$ .

Identify the known angle and sides relative to $y$ .

Solve for $y$ .

Identify the known angle and sides relative to $z$ .

Solve for $z$ .