Question

find the sine, cosine, and tangent of \\( \angle i \\).
simplify your answers and write them as proper fractions, improper fractions, or whole
numbers.
\\( \sin (i)= \\)
\\( \cos (i)= \\)
\\( \tan (i)= \\)

Explanation:

Step1: Find the length of \( JK \)

Use the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \). Let \( a = 36 \), \( c = 85 \), and \( b=JK \).

$$
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Step2: Calculate \( \sin(I) \)

Recall the formula \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle I \), the opposite side to \( \angle I \) is \( JK = 77 \), and the hypotenuse is \( 85 \).

$$ \sin(I)=\frac{77}{85} $$

Step3: Calculate \( \cos(I) \)

Recall the formula \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \angle I \), the adjacent side to \( \angle I \) is \( IK = 36 \), and the hypotenuse is \( 85 \).

$$ \cos(I)=\frac{36}{85} $$

Step4: Calculate \( \tan(I) \)

Recall the formula \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle I \), the opposite side to \( \angle I \) is \( JK = 77 \), and the adjacent side is \( IK = 36 \).

$$ \tan(I)=\frac{77}{36} $$

Answer:

\(\sin(I)=\frac{77}{85}\), \(\cos(I)=\frac{36}{85}\), \(\tan(I)=\frac{77}{36}\)