Question

the point given below is on the terminal side of an angle θ. find the exact value of each of the six trigonometric functions of θ. (-24, 7)

Explanation:

Step1: Calculate the radius $r$

Use the formula $r=\sqrt{x^{2}+y^{2}}$, where $x = - 24$ and $y = 7$. So $r=\sqrt{(-24)^{2}+7^{2}}=\sqrt{576 + 49}=\sqrt{625}=25$.

Step2: Calculate $\sin\theta$

By the definition $\sin\theta=\frac{y}{r}$, substituting $y = 7$ and $r = 25$, we get $\sin\theta=\frac{7}{25}$.

Step3: Calculate $\cos\theta$

By the definition $\cos\theta=\frac{x}{r}$, substituting $x=-24$ and $r = 25$, we get $\cos\theta=-\frac{24}{25}$.

Step4: Calculate $\tan\theta$

By the definition $\tan\theta=\frac{y}{x}$, substituting $x=-24$ and $y = 7$, we get $\tan\theta=-\frac{7}{24}$.

Step5: Calculate $\csc\theta$

Since $\csc\theta=\frac{r}{y}$, substituting $y = 7$ and $r = 25$, we get $\csc\theta=\frac{25}{7}$.

Step6: Calculate $\sec\theta$

Since $\sec\theta=\frac{r}{x}$, substituting $x=-24$ and $r = 25$, we get $\sec\theta=-\frac{25}{24}$.

Step7: Calculate $\cot\theta$

Since $\cot\theta=\frac{x}{y}$, substituting $x=-24$ and $y = 7$, we get $\cot\theta=-\frac{24}{7}$.

Answer:

$\sin\theta=\frac{7}{25}$, $\cos\theta=-\frac{24}{25}$, $\tan\theta=-\frac{7}{24}$, $\csc\theta=\frac{25}{7}$, $\sec\theta=-\frac{25}{24}$, $\cot\theta=-\frac{24}{7}$