Question

refer to right triangle abc with as shown in the figure (not drawn to scale).
use the given information to find the six trigonometric functions of a. answer exactly.
a = 3, b = 2
sin(a) =
cos(a) =
tan(a) =
cot(a) =
sec(a) =
csc(a) =
question help: video ebook written example message instructor

Explanation:

Step1: Find the hypotenuse \(c\)

By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\). Given \(a = 3\) and \(b=2\), then \(c=\sqrt{3^{2}+2^{2}}=\sqrt{9 + 4}=\sqrt{13}\).

Step2: Calculate \(\sin(A)\)

\(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}\). Substituting \(a = 3\) and \(c=\sqrt{13}\), we get \(\sin(A)=\frac{3}{\sqrt{13}}=\frac{3\sqrt{13}}{13}\).

Step3: Calculate \(\cos(A)\)

\(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}\). Substituting \(b = 2\) and \(c=\sqrt{13}\), we get \(\cos(A)=\frac{2}{\sqrt{13}}=\frac{2\sqrt{13}}{13}\).

Step4: Calculate \(\tan(A)\)

\(\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}\). Substituting \(a = 3\) and \(b = 2\), we get \(\tan(A)=\frac{3}{2}\).

Step5: Calculate \(\cot(A)\)

\(\cot(A)=\frac{\text{adjacent}}{\text{opposite}}=\frac{b}{a}\). Substituting \(a = 3\) and \(b = 2\), we get \(\cot(A)=\frac{2}{3}\).

Step6: Calculate \(\sec(A)\)

\(\sec(A)=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{c}{b}\). Substituting \(b = 2\) and \(c=\sqrt{13}\), we get \(\sec(A)=\frac{\sqrt{13}}{2}\).

Step7: Calculate \(\csc(A)\)

\(\csc(A)=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{c}{a}\). Substituting \(a = 3\) and \(c=\sqrt{13}\), we get \(\csc(A)=\frac{\sqrt{13}}{3}\).

Answer:

\(\sin(A)=\frac{3\sqrt{13}}{13}\), \(\cos(A)=\frac{2\sqrt{13}}{13}\), \(\tan(A)=\frac{3}{2}\), \(\cot(A)=\frac{2}{3}\), \(\sec(A)=\frac{\sqrt{13}}{2}\), \(\csc(A)=\frac{\sqrt{13}}{3}\)