Question

use the pythagorean theorem to find the length of the missing side of the right triangle. then find the value of each of the six trigonometric functions of θ.
the length of the missing side of the right triangle is a = 3.
sin θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cos θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
tan θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
csc θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
sec θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cot θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Apply Pythagorean Theorem

According to the Pythagorean Theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 25\) and \(b = 20\). We solve for \(a\): \(a=\sqrt{c^{2}-b^{2}}=\sqrt{25^{2}-20^{2}}=\sqrt{(25 + 20)(25 - 20)}=\sqrt{45\times5}=\sqrt{225}=15\) (the value of \(a = 3\) in the problem is incorrect).

Step2: Define sine function

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}=\frac{15}{25}=\frac{3}{5}\)

Step3: Define cosine function

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}=\frac{20}{25}=\frac{4}{5}\)

Step4: Define tangent function

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{15}{20}=\frac{3}{4}\)

Step5: Define cosecant function

\(\csc\theta=\frac{1}{\sin\theta}=\frac{5}{3}\)

Step6: Define secant function

\(\sec\theta=\frac{1}{\cos\theta}=\frac{5}{4}\)

Step7: Define cotangent function

\(\cot\theta=\frac{1}{\tan\theta}=\frac{4}{3}\)

Answer:

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