Answer:
[tex]\displaystyle cos\theta=-\frac{\sqrt{5}}{3}[/tex]
[tex]\displaystyle tan\theta=-\frac{2\sqrt{5}}{5}[/tex]
Step-by-step explanation:
Trigonometric Formulas
To solve this problem, we must recall some basic relations and concepts.
The main trigonometric identity relates the sine to the cosine:
[tex]sin^2\theta+cos^2\theta=1[/tex]
The tangent can be found by
[tex]\displaystyle tan\theta=\frac{sin\theta}{cos\theta}[/tex]
The cosine and the secant are related by
[tex]\displaystyle cos\theta=\frac{1}{sec\theta}[/tex]
They both have the same sign.
The sine is positive in the first and second quadrants, the cosine is positive in the first and fourth quadrants.
The sine is negative in the third and fourth quadrants, the cosine is negative in the second and third quadrants.
We are given
[tex]\displaystyle sin\theta=\frac{2}{3}[/tex]
Find the cosine by solving
[tex]sin^2\theta+cos^2\theta=1[/tex]
[tex]\displaystyle \left(\frac{2}{3}\right)^2+cos^2\theta=1[/tex]
[tex]\displaystyle cos^2\theta=1-\left(\frac{2}{3}\right)^2=1-\frac{4}{9}=\frac{5}{9}[/tex]
[tex]\displaystyle cos\theta=\sqrt{\frac{5}{9}}=-\frac{\sqrt{5}}{3}[/tex]
[tex]\boxed{\displaystyle cos\theta=-\frac{\sqrt{5}}{3}}[/tex]
We have placed the negative sign because we know the secant ('sex') is negative and they both have the same sign.
Now compute the tangent
[tex]\displaystyle tan\theta=\frac{sin\theta}{cos\theta}=\frac{\frac{2}{3}}{-\frac{\sqrt{5}}{3}}=-\frac{2}{\sqrt{5}}[/tex]
Rationalizing
[tex]\displaystyle tan\theta=-\frac{2}{\sqrt{5}}=-\frac{2\sqrt{5}}{5}[/tex]
[tex]\boxed{\displaystyle tan\theta=-\frac{2\sqrt{5}}{5}}[/tex]
