Question

find tan j and tan k. write each answer as a fraction in simplest form and as a decimal rounded to four places.
(there is a right triangle jlk with right angle at l. side jl is 32, side lk is 24, side jk is 40. then there are two parts: tan j =
=
≈
and tan k =
≈
)

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( J \), the opposite side is \( LK = 24 \) and the adjacent side is \( JL = 32 \). For angle \( K \), the opposite side is \( JL = 32 \) and the adjacent side is \( LK = 24 \).

Step2: Calculate \( \tan J \)

Using the definition of tangent, \( \tan J=\frac{\text{opposite to } J}{\text{adjacent to } J}=\frac{LK}{JL} \). Substituting the values \( LK = 24 \) and \( JL = 32 \), we get \( \tan J=\frac{24}{32} \). Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor (which is 8), we have \( \frac{24\div8}{32\div8}=\frac{3}{4} \). As a decimal, \( \frac{3}{4}=0.75 \).

Step3: Calculate \( \tan K \)

Using the definition of tangent, \( \tan K=\frac{\text{opposite to } K}{\text{adjacent to } K}=\frac{JL}{LK} \). Substituting the values \( JL = 32 \) and \( LK = 24 \), we get \( \tan K=\frac{32}{24} \). Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor (which is 8), we have \( \frac{32\div8}{24\div8}=\frac{4}{3} \approx 1.3333 \) (rounded to four decimal places).

Answer:

For \( \tan J \):

• As a fraction: \( \frac{3}{4} \)
• As a decimal: \( 0.75 \)

For \( \tan K \):

• As a fraction: \( \frac{4}{3} \)
• As a decimal: \( \approx 1.3333 \)