Question

find the six trigonometric function values of the specified angle.
sin θ = 7√74 / 74 (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.)
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

Explanation:

Step1: Find hypotenuse

By Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}=\sqrt{5^{2} + 12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13$.

Step2: Calculate sine

$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{5}{13}$.

Step3: Calculate cosine

$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{13}$.

Step4: Calculate tangent

$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{12}$.

Step5: Calculate cosecant

$\csc(\theta)=\frac{1}{\sin(\theta)}=\frac{13}{5}$.

Step6: Calculate secant

$\sec(\theta)=\frac{1}{\cos(\theta)}=\frac{13}{12}$.

Step7: Calculate cotangent

$\cot(\theta)=\frac{1}{\tan(\theta)}=\frac{12}{5}$.

Answer:

$\sin(\theta)=\frac{5}{13}$, $\cos(\theta)=\frac{12}{13}$, $\tan(\theta)=\frac{5}{12}$, $\csc(\theta)=\frac{13}{5}$, $\sec(\theta)=\frac{13}{12}$, $\cot(\theta)=\frac{12}{5}$