3. Trigonometry
c. Circle Definitions
Recall: \[\begin{aligned} \sin\theta&=\dfrac{y}{r}\quad &\tan\theta&=\dfrac{y}{x}\quad &\sec\theta&=\dfrac{r}{x} \\[8pt] \cos\theta&=\dfrac{x}{r}\quad &\cot\theta&=\dfrac{x}{y}\quad &\csc\theta&=\dfrac{r}{y} \end{aligned}\]
2. Angles Bigger than \(90^\circ\)
When the angle is bigger than \(\dfrac{\pi}{2}\), we still use the same formulas for the trig functions, listed above.
Consider a circle centered at the origin and the ray at the angle \(\theta\) which passes through the point \((-12,35)\), find the radius of the circle and identify the six trig functions for the angle \(\theta\).
![eg_12-35-37](graphics/eg_12-35-37.jpg)
Since the ray at the angle \(\theta\) passes through the point \((x,y)=(-12,35)\), the radius is \[ r=\sqrt{12^2+35^2}=\sqrt{144+1225}=\sqrt{1369}=37 \] So the trig functions are: \[\begin{aligned} \sin\theta&=\dfrac{y}{r}=\dfrac{35}{37}\quad &\tan\theta&=\dfrac{y}{x}=-\,\dfrac{35}{12}\quad &\sec\theta&=\dfrac{r}{x}=-\,\dfrac{37}{12} \\[8pt] \cos\theta&=\dfrac{x}{r}=-\,\dfrac{12}{37}\quad &\cot\theta&=\dfrac{x}{y}=-\,\dfrac{12}{35}\quad &\csc\theta&=\dfrac{r}{y}=\dfrac{37}{35} \end{aligned}\]
Consider a circle centered at the origin and the ray at the angle \(\theta\) which passes through the point \((-8,-15)\), find the radius of the circle and identify the six trig functions for the angle \(\theta\).
![ex_8-15-17](graphics/ex_8-15-17.jpg)
\(r=17\)
\(
\sin\theta=-\,\dfrac{15}{17}\quad
\tan\theta=\dfrac{15}{8}\quad
\sec\theta=-\,\dfrac{17}{8}
\)
\(
\cos\theta=-\,\dfrac{8}{17}\quad
\cot\theta=\dfrac{8}{15}\quad
\csc\theta=-\,\dfrac{17}{15}
\)
Since the ray at the angle \(\theta\) passes through the point \((x,y)=(-8,-15)\), the radius is \[ r=\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17 \] So the trig functions are: \[\begin{aligned} \sin\theta&=\dfrac{y}{r}=-\,\dfrac{15}{17}\quad &\tan\theta&=\dfrac{y}{x}=\dfrac{15}{8}\quad &\sec\theta&=\dfrac{r}{x}=-\,\dfrac{17}{8} \\[8pt] \cos\theta&=\dfrac{x}{r}=-\,\dfrac{8}{17}\quad &\cot\theta&=\dfrac{x}{y}=\dfrac{8}{15}\quad &\csc\theta&=\dfrac{r}{y}=-\,\dfrac{17}{15} \end{aligned}\]
Consider a circle centered at the origin and the ray at the angle \(\theta\) which passes through the point \((7,-24)\), find the radius of the circle and identify the six trig functions for the angle \(\theta\).
![ex_7-24-25](graphics/ex_7-24-25.jpg)
\(r=25\)
\(
\sin\theta=-\,\dfrac{24}{25}\quad\,
\tan\theta=-\,\dfrac{24}{7}\quad
\sec\theta=\dfrac{25}{7}
\)
\(
\cos\theta=\dfrac{7}{25}\qquad
\cot\theta=-\,\dfrac{7}{24}\quad
\csc\theta=-\,\dfrac{25}{24}
\)
Since the ray at the angle \(\theta\) passes through the point \((x,y)=(7,-24)\), the radius is \[ r=\sqrt{7^2+24^2}=\sqrt{49+576}=\sqrt{625}=25 \] So the trig functions are: \[\begin{aligned} \sin\theta&=\dfrac{y}{r}=-\,\dfrac{24}{25}\quad &\tan\theta&=\dfrac{y}{x}=-\,\dfrac{24}{7}\quad &\sec\theta&=\dfrac{r}{x}=\dfrac{25}{7} \\[8pt] \cos\theta&=\dfrac{x}{r}=\dfrac{7}{25}\quad &\cot\theta&=\dfrac{x}{y}=-\,\dfrac{7}{24}\quad &\csc\theta&=\dfrac{r}{y}=-\,\dfrac{25}{24} \end{aligned}\]
Heading
Placeholder text: Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum