Chapter 9: Problem 16

(a) Sketch $y=\cos \left(x-20^{\circ}\right)$, $0^{\circ} \leq x \leq 360^{\circ}$ (b) On the same axes, sketch $y=\sin x$. (c) Use your graphs to obtain approximate solutions of $$ \sin x=\cos \left(x-20^{\circ}\right) $$

Short Answer

Expert verified

Answer: The approximate solutions for the equation are x≈55° and x≈245°.

Step by step solution

Sketch y=cos(x-20°)

First, we will sketch the graph of the function $y=\cos(x-20^{\circ})$ in the domain of $0^{\circ}\leq x \leq 360^{\circ}$. To do this, we need to plot a few important points: - The maximum value of the cosine function is 1, which corresponds to $x=20^{\circ}$. - The minimum value of the cosine function is -1, which corresponds to $x=200^{\circ}$. - The function has zero values for $x=110^{\circ}$ and $x=290^{\circ}$. The graph starts from the maximum value of 1, goes to zero at $x=110^{\circ}$, reaches the minimum of -1 at $x=200^{\circ}$, and finally returns to zero at $x=290^{\circ}$ and to 1 at the end of the domain.

Sketch y=sin(x)

Now we will sketch the graph of the function $y=\sin x$ in the same domain of $0^{\circ}\leq x \leq 360^{\circ}$. Again, we need to plot a few important points: - The maximum value of the sine function is 1, which corresponds to $x=90^{\circ}$. - The minimum value of the sine function is -1, which corresponds to $x=270^{\circ}$. - The function has zero values for $x=0^{\circ}, 180^{\circ}$, and $360^{\circ}$. The graph starts from zero, goes to the maximum value of 1 at $x=90^{\circ}$, goes back to zero at $x=180^{\circ}$, reaches the minimum of -1 at $x=270^{\circ}$, and finally returns to zero at the end of the domain.

Find approximate solutions of sin(x)=cos(x-20°) using the graphs

To find the approximate solutions of the equation $\sin x=\cos(x-20^{\circ})$, we need to find the points where the two graphs intersect. By analyzing the graphs, we notice that the intersection points occur at two locations in the domain: 1. The first intersection occurs between $x=50^{\circ}$ and $x=60^{\circ}$. We can estimate this intersection as: $x_1\approx55^{\circ}$. 2. The second intersection occurs between $x=240^{\circ}$ and $x=250^{\circ}$. We can estimate this intersection as: $x_2\approx245^{\circ}$. Therefore, the approximate solutions of the equation are $x\approx55^{\circ}$ and $x\approx245^{\circ}$.

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(a) Sketch \(y=\cos \left(x-20^{\circ}\right)\), \(0^{\circ} \leq x \leq 360^{\circ}\) (b) On the same axes, sketch \(y=\sin x\). (c) Use your graphs to obtain approximate solutions of $$ \sin x=\cos \left(x-20^{\circ}\right) $$

Short Answer

Step by step solution

Sketch y=cos(x-20°)

Sketch y=sin(x)

Find approximate solutions of sin(x)=cos(x-20°) using the graphs

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