Question

a point on the terminal side of an angle ( \theta ) in standard position is ( (3,-2) ). find the exact value of each of the six trigonometric functions of ( \theta ).
select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. ( sin \theta=square )
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function is not defined.
select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. ( cos \theta=square )
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function is not defined.
select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. ( \tan \theta=square )
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function is not defined.
select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. ( csc \theta=square )
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function is not defined,
select the correct choice and, if necessary, fill in the answer box to complete your choice

Explanation:

Step1: Calculate the value of \(r\)

For a point \((x,y)\) on the terminal side of an angle \(\theta\), \(r=\sqrt{x^{2}+y^{2}}\). Here \(x = 3\) and \(y=-2\), so \(r=\sqrt{3^{2}+(-2)^{2}}=\sqrt{9 + 4}=\sqrt{13}\)

Step2: Calculate \(\sin\theta\)

The formula for \(\sin\theta=\frac{y}{r}\). Substituting \(y=-2\) and \(r = \sqrt{13}\), we get \(\sin\theta=\frac{-2}{\sqrt{13}}=-\frac{2\sqrt{13}}{13}\)

Step3: Calculate \(\cos\theta\)

The formula for \(\cos\theta=\frac{x}{r}\). Substituting \(x = 3\) and \(r=\sqrt{13}\), we get \(\cos\theta=\frac{3}{\sqrt{13}}=\frac{3\sqrt{13}}{13}\)

Step4: Calculate \(\tan\theta\)

The formula for \(\tan\theta=\frac{y}{x}\). Substituting \(x = 3\) and \(y=-2\), we get \(\tan\theta=\frac{-2}{3}=-\frac{2}{3}\)

Step5: Calculate \(\csc\theta\)

Since \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin\theta=-\frac{2\sqrt{13}}{13}\), then \(\csc\theta=-\frac{\sqrt{13}}{2}\)

Answer:

• \(\sin\theta=-\frac{2\sqrt{13}}{13}\)
• \(\cos\theta=\frac{3\sqrt{13}}{13}\)
• \(\tan\theta=-\frac{2}{3}\)
• \(\csc\theta=-\frac{\sqrt{13}}{2}\)