Question

find the exact values of the six trigonometric functions of each angle θ. (a) image of coordinate plane with point (4,3) and angle θ sin(θ) =
cos(θ) =
tan(θ) =
csc(θ) =
sec(θ) =
cot(θ) =

Explanation:

Step1: Determine r (hypotenuse)

Given the point $(4, 3)$, we use the formula $r = \sqrt{x^2 + y^2}$. Here, $x = 4$ and $y = 3$. So, $r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.

Step2: Calculate $\sin(\theta)$

Using the definition $\sin(\theta) = \frac{y}{r}$, substitute $y = 3$ and $r = 5$. So, $\sin(\theta) = \frac{3}{5}$.

Step3: Calculate $\cos(\theta)$

Using the definition $\cos(\theta) = \frac{x}{r}$, substitute $x = 4$ and $r = 5$. So, $\cos(\theta) = \frac{4}{5}$.

Step4: Calculate $\tan(\theta)$

Using the definition $\tan(\theta) = \frac{y}{x}$, substitute $y = 3$ and $x = 4$. So, $\tan(\theta) = \frac{3}{4}$.

Step5: Calculate $\csc(\theta)$

Using the reciprocal of $\sin(\theta)$, $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{r}{y}$. Substitute $r = 5$ and $y = 3$. So, $\csc(\theta) = \frac{5}{3}$.

Step6: Calculate $\sec(\theta)$

Using the reciprocal of $\cos(\theta)$, $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{r}{x}$. Substitute $r = 5$ and $x = 4$. So, $\sec(\theta) = \frac{5}{4}$.

Step7: Calculate $\cot(\theta)$

Using the reciprocal of $\tan(\theta)$, $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{x}{y}$. Substitute $x = 4$ and $y = 3$. So, $\cot(\theta) = \frac{4}{3}$.

Answer:

$\sin(\theta)=\frac{3}{5}$, $\cos(\theta)=\frac{4}{5}$, $\tan(\theta)=\frac{3}{4}$, $\csc(\theta)=\frac{5}{3}$, $\sec(\theta)=\frac{5}{4}$, $\cot(\theta)=\frac{4}{3}$