Question

10 - 12. what is the value of x in each diagram below? homework help
a.
triangle with sides 17, x and angles 50°, 20°
b.
triangle with sides 10, x and angles 55°, 36°
10 - 13. write a possible equation for each graph. homework help
a.
graph a
b.
graph b
c.
graph c
d.
graph d

Explanation:

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10-12: Solve for $x$ in each triangle

We use the Law of Sines for both triangles, which states $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, where $a$ is the side opposite angle $A$, etc.

Part a:

Step1: Find third angle

The third angle of the triangle is $180^\circ - 50^\circ - 20^\circ = 110^\circ$.

Step2: Apply Law of Sines

Set up the ratio:
$$\frac{x}{\sin 110^\circ} = \frac{17}{\sin 20^\circ}$$
Solve for $x$:
$$x = \frac{17 \sin 110^\circ}{\sin 20^\circ}$$
Calculate values: $\sin 110^\circ \approx 0.9397$, $\sin 20^\circ \approx 0.3420$
$$x \approx \frac{17 \times 0.9397}{0.3420} \approx 46.3$$

Part b:

Step1: Find third angle

The third angle of the triangle is $180^\circ - 55^\circ - 36^\circ = 89^\circ$.

Step2: Apply Law of Sines

Set up the ratio:
$$\frac{x}{\sin 55^\circ} = \frac{10}{\sin 89^\circ}$$
Solve for $x$:
$$x = \frac{10 \sin 55^\circ}{\sin 89^\circ}$$
Calculate values: $\sin 55^\circ \approx 0.8192$, $\sin 89^\circ \approx 0.9998$
$$x \approx \frac{10 \times 0.8192}{0.9998} \approx 8.2$$

Graph a:

Step1: Identify key features

Amplitude $A=1$, Period $=2\pi$, Vertical shift $D=1$, No horizontal shift, starts at $(0,1)$ (cosine shape).

Step2: Write the equation

$$y = \cos(x) + 1$$

Graph b:

Step1: Identify key features

Amplitude $A=1$, Period $=2\pi$, Vertical shift $D=1$, Horizontal shift $C=0$, starts at $(0,2)$ (reflected cosine shape).

Step2: Write the equation

$$y = -\cos(x) + 1$$

Graph c:

Step1: Identify key features

Amplitude $A=1$, Period $=2\pi$, Vertical shift $D=1$, Horizontal shift $C=0$, starts at $(0,1)$ (sine shape shifted up).

Step2: Write the equation

$$y = \sin(x) + 1$$

Graph d:

Step1: Identify key features

Amplitude $A=1$, Period $=2\pi$, Vertical shift $D=-1$, Horizontal shift $C=0$, starts at $(0,-2)$ (reflected cosine shape shifted down).

Step2: Write the equation

$$y = -\cos(x) - 1$$

Answer:

(10-12):
a. $x \approx 46.3$
b. $x \approx 8.2$

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10-13: Write equations for each trigonometric graph

We use the general sinusoidal form: $y = A\sin(B(x-C)) + D$ or $y = A\cos(B(x-C)) + D$, where:

• $A$ = amplitude, $B = \frac{2\pi}{\text{period}}$, $C$ = horizontal shift, $D$ = vertical shift