Respuesta :
[tex]\begin{gathered} \sin \text{ }\theta\text{ = }\frac{2\sqrt[]{5}}{5} \\ \\ \cos \text{ }\theta\text{ = -}\frac{\sqrt[]{5}}{5} \\ \\ \text{Tan }\theta\text{ = -2} \\ \\ co\sec \text{ }\theta\text{ = }\frac{\sqrt[]{5}}{2} \\ \\ \sec \text{ }\theta\text{ = -}\sqrt[]{5} \\ \\ \cot \text{ }\theta\text{ = -}\frac{1}{2} \end{gathered}[/tex]
Firstly, we will have to make a representation of the angle
We have this as follows;
As we can see, alpha added to theta is 180 degrees
Firstly, by the use of Pythagoras' theorem, we can get the value of r
r faces the right angle, and that makes it the hypotenuse
According to the theorem, the square of r, the hypotenuse equals the sum of the squares of the two other sides
Thus, we have it that;
[tex]\begin{gathered} r^2=(-2)^2+4^2 \\ r^2\text{ = 4 + 16} \\ r^2\text{ = 20} \\ r\text{ = }\sqrt[]{20} \\ r\text{ = 2}\sqrt[]{5} \end{gathered}[/tex]From here, we can proceed to get the individual trigonometric ratios
a) Sine
This is the ratio of the opposite to the hypotenuse
On the second quadrant, the value of sine is positive
Thus, we have it that;
[tex]\begin{gathered} \sin \text{ }\alpha\text{ = }\frac{4}{2\sqrt[]{5}} \\ \alpha\text{ = }\sin ^{-1}(\frac{4}{2\sqrt[]{5}}) \\ \alpha\text{ = 63.43} \\ \theta\text{ = 180-63.43} \\ \theta\text{ = 116.57} \\ \sin \text{ 116.57 = }\frac{4}{2\sqrt[]{5}\text{ }}=\text{ }\frac{4\sqrt[]{5}}{10}\text{ = }\frac{2\sqrt[]{5}}{5} \\ \\ \sin \text{ }\theta\text{ = }\frac{2\sqrt[]{5}}{5} \end{gathered}[/tex]b) cosine
The cosine of an angle is the ratio of the adjacent to the hypotenuse
Mathematically, we know that;
[tex]\begin{gathered} \cos ^2\theta+sin^2\theta\text{ = 1} \\ \cos ^2\theta=1-sin^2\theta \\ \cos ^2\theta\text{ = 1 - (}\frac{2\sqrt[]{5}}{5})^2 \\ \\ \cos ^2\theta\text{ = 1- }\frac{20}{25} \\ \\ \cos ^2\theta\text{ = }\frac{5}{25} \\ \cos ^2\theta\text{ = }\frac{1}{5} \\ \\ \cos \text{ }\theta\text{ = }\sqrt[]{\frac{1}{5}} \\ \\ \cos \text{ }\theta\text{ = -}\frac{\sqrt[]{5}}{5} \end{gathered}[/tex]We choose the negative value for the cosine since cosine is negative on the second quadrant
c) Tan
The tan of an angle is the ratio of the opposite to the adjacent
Also, by dividing the sine of an angle by the cosine of the same angle, we can get the tan of the angle
Thus, we have it that;
[tex]\begin{gathered} \text{Tan }\theta\text{ = }\frac{\sin \text{ }\theta}{\cos \text{ }\theta} \\ \\ \text{Tan }\theta\text{ = }\frac{\frac{2\sqrt[]{5}}{5}}{\frac{-\sqrt[]{5}}{5}}\text{ = }\frac{2\sqrt[]{5}}{5}\times\frac{5}{-\sqrt[]{5}}\text{ = -2} \end{gathered}[/tex]d) cosec theta
The cosec of an angle is the multiplicative inverse of the sine
Mathematically;
[tex]\begin{gathered} co\sec \theta\text{ = }\frac{1}{\sin \text{ }\theta} \\ \\ co\sec \text{ }\theta\text{ = }\frac{1}{\frac{2\sqrt[]{5}}{5}}\text{ = }\frac{5}{2\sqrt[]{5}}\text{ = }\frac{5\sqrt[]{5}}{10}\text{ = }\frac{\sqrt[]{5}}{2} \end{gathered}[/tex]e) sec theta
The sec of an angle is the multiplicative inverse of the cosine of the angle
Thus, we have it that;
[tex]\text{sec }\theta\text{ = }\frac{1}{\cos \text{ }\theta}\text{ = }\frac{1}{-\frac{\sqrt[]{5}}{5}}\text{ = -}\frac{5}{\sqrt[]{5}}\text{ = -}\frac{5\sqrt[]{5}}{5}\text{ = -}\sqrt[]{5}[/tex]f) cot theta
The cot of an angle is the multiplicative angle of the tan
Thus, we have it that;
[tex]\begin{gathered} \cot \text{ }\theta\text{ = }\frac{1}{\tan \text{ }\theta} \\ \\ \cot \text{ }\theta\text{ = }\frac{1}{-2}\text{ = -}\frac{1}{2} \end{gathered}[/tex]
