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find the six trigonometric function values of the specified angle.
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.)
answer: assuming the angle θ is opposite the side with length 5, the values are:
· sin(θ) = 5/13
· cos(θ) = 12/13
· tan(θ) = 5/12
· csc(θ) = 13/5
· sec(θ) = 13/12
· cot(θ) = 12/5
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Explanation:

Step1: Find the hypotenuse

By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 5\) and \(b=7\). So \(c=\sqrt{5^{2}+7^{2}}=\sqrt{25 + 49}=\sqrt{74}\).

Step2: Calculate sine

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{5}{\sqrt{74}}=\frac{5\sqrt{74}}{74}\)

Step3: Calculate cosine

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{7}{\sqrt{74}}=\frac{7\sqrt{74}}{74}\)

Step4: Calculate tangent

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{7}\)

Step5: Calculate cosecant

\(\csc\theta=\frac{1}{\sin\theta}=\frac{\sqrt{74}}{5}\)

Step6: Calculate secant

\(\sec\theta=\frac{1}{\cos\theta}=\frac{\sqrt{74}}{7}\)

Step7: Calculate cotangent

\(\cot\theta=\frac{1}{\tan\theta}=\frac{7}{5}\)

Answer:

\(\sin\theta=\frac{5\sqrt{74}}{74}\), \(\cos\theta=\frac{7\sqrt{74}}{74}\), \(\tan\theta=\frac{5}{7}\), \(\csc\theta=\frac{\sqrt{74}}{5}\), \(\sec\theta=\frac{\sqrt{74}}{7}\), \(\cot\theta=\frac{7}{5}\)